- Email: [email protected]
Thermodynamic framework for work-assisted unit operations
Exergy Int. J. 1(3) (2001) 166–172 www.exergyonline.com
Thermodynamic framework for work-assisted unit operations Stanislaw Sieniutycz ∗ , Zbigniew Szwast Faculty of Chemical Engineering, Warsaw University of Technology, 1 Warynskiego Street, 00-645 Warsaw, Poland
(Received 16 July 2000, accepted 6 February 2001)
Abstract — We construct thermodynamic criteria for optimization of sequential work-assisted unit operations, especially for heating, evaporation and drying. Constraints take into account dynamics of heat and mass transport and rate of real work consumption. Finiterate, endoreversible models are used. By definition, they refer to systems which have separated in space their internal reversible parts, such as Carnot engines or heat pumps, from their (external) purely irreversible parts, such as thermal resistances. Consequently these models can quantify irreducible losses of classical exergy caused by the resistances. Extremum performance functions for optimal work, which incorporate residual minimum entropy production, are determined in terms of end states, duration and (in discrete processes) number of stages. Analogies between entropy production expressions for work-assisted operations and those without work help to formulate optimization models of the former. Our theory also answers the question whether work production is at all possible in a given system. 2001 Éditions scientifiques et médicales Elsevier SAS
Nomenclature A bg , bs
available energy (exergy) . . . specific exergy of gas and gas in equilibrium with solid . . . . . b specific exergy of controlling phase . . . . . . . . . . . . . . c specific heat at the constant pressure . . . . . . . . . . . . . G mass flux, total flow rate . . . partial and overall conductances g1 , g HTU height of transfer uni . . . . . solid enthalpy at stage n . . . . In i specific enthalpy of controlling phase . . . . . . . . . . . . . . ig , is specific enthalpy of gas and gas at equilibrium with solid . . . N total number of stages in a multistage process n current stage number of a multistage process P n , pn cumulative power output and power output at nth stage . . .
q1
kJ
kJ·kg−1 kJ·kg−1 kJ·K−1 ·kg−1 kg·s kJ·K−1 ·s−1 m kJ·kg−1 kJ·kg−1 kJ·kg−1
kJ·s−1
∗ Correspondence and reprints.
E-mail address: [email protected] (S. Sieniutycz).
166
driving heat in the engine mode of the stage . . . . . . . . . . . kJ·s−1 n R (x, t) optimal work function of cost type in terms of state and time kJ·kg−1 S solid entropy, entropy of controlled phase . . . . . . . . . . kJ·K−1 s specific entropy of controlling phase, (gas) . . . . . . . . . . . kJ·K−1 ·kg−1 Sσ specific entropy production . . kJ·kg−1 ·K−1 ·s−1 T temperature of controlled phase (solid) . . . . . . . . . . . . . . K Te constant equilibrium temperature of reservoir . . . . . . . . . . . K T temperature of controlling phase (gas) . . . . . . . . . . . . . . K t physical time, contact time . . s un = T n /τ n rate of the temperature change as the control variable . . . . . . . . K V ≡ max W optimal work function of profit type . . . . . . . . . . kJ·K−1 W ≡ P /G total specific work or total power per unit mass flux . kJ·K−1 n W moisture content in solid from stage n . . . . . . . . . . . . . kg−1 ·kg−1 X absolute humidity of controlling phase (gas) . . . . . . . . . . . kg−1 ·kg−1 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1164-0235(01)00022-X/FLA
S. Sieniutycz, Z. Szwast / Exergy Int. J. 1(3) (2001) 166–172
x α η θn
transfer area coordinate . . . . overall heat transfer coefficient = p/q1 first-law efficiency free interval of an independent variable or nondimensional time interval at stage n chemical potential of kth component . . . . . . . . . . . coefficient of outlet gas utilization nondimensional time, number of the heat transfer units (x/HTU )
µk µ τ
m kJ·m−2 ·K−1 ·s−1
kJ·kg−1
Subscripts g i s 1, 2
gas ith state variable saturation, solid first and second fluid
Superscrits e f i k
environment, equilibrium final state initial state or n number of kth or nth stage
1. INTRODUCTION This work analyzes thermodynamic criteria for optimization of work-assisted operations and compares these criteria with those for conventional operations (without work). Basically we display here a quantitative analysis for power criteria associated with production or consumption of mechanical work and for the criterion of entropy production; it is one of the main objectives of this work to show how all these criteria can be reconciled in the realm of thermal machines and traditional exchangers (without work). Our work contributes to the quantitative theory of work production and consumption in single-stage and multi-stage thermal machines in which the effect of fluxes on efficiencies is essential. For sequential operations with thermal machines, such as multistage heat pumps, total power input is minimized at constraints which describe dynamics of energy and mass exchange. The results are optimal work functions in terms of end states, duration and (in discrete processes) number of stages. Modeling a general work-assisted operation is a difficult task. However, formal analogies do exist between entropy production expressions in workassisted and in conventional operations. These analogies are helpful to develop suitable criteria and models. We shall also answer the question whether work production is at all possible in a given system. This problem is not
completely solved in the contemporary theory of thermal machines based on the finite-time thermodynamics (FTT; Refs. [1–3], to which the reader is referred for an extensive literature review of earlier issues). To begin with we consider the heat transfer-driven work generation (consumption) in an ‘endoreversible’ thermal machine, an engine (figure 1) or heat pump, which interacts with a high-T fluid (e.g., drying gas) flowing with the mass flux Gf [4, 5]. The multistage production (consumption) of work requires to use the sequence of Novikov–Chambadal–Curzon–Ahlborn (NCCA) stages (figure 2). In an endoreversible engine a resource fluid drives the Carnot engine from which the work is taken out, in an endoreversible heat pump a fluid (e.g., drying agent) is driven in the condenser of the Carnot heat pump to which work is supplied; in both cases the second fluid is an infinite reservoir. The fluids are of finite thermal conductivity, hence there are finite thermal resistances in the system. In a multistage heating operation the fluid’s temperature increases at each stage; the whole operation is described by the sequence T 0 , T 1 , . . . , T N . Superscripts refer to the stage number; this rule includes also constant process parameters and initial and final values of state variables of a limiting continuous process as they are respectively inlet and outlet
Figure 1. A single-stage NCCA engine as an operation with active heat exchange between two fluids (in the engine mode T1 0).
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S. Sieniutycz, Z. Szwast / Exergy Int. J. 1(3) (2001) 166–172
is q1 = −Gf c(T 1 − T 0 ), where Gf is the fluid flow and c-fluid’s specific heat. The specific work produced in a single endoreversible engine or that consumed in a single endoreversible heat pump is [4, 5] q1 Te (1) W ≡ −p/Gf = 1 − T − q1 /g Gf where the bracketed expression is the first-law efficiency. Here p is the power output, g—an overall thermal conductance of thermal machine related to an overall heat transfer coefficient, α . In multistage systems (figure 2) one should sum expressions such as equation (1) over stages. Casting the problem in the format of a discrete maximum principle we arrive at the discrete functional of consumed work N −W N = c 1− n=1
Figure 2. A multi-stage NCCA engine as a sequential operation with many Carnot loops to accomplish active heat exchange between two fluids (in the engine mode T1 0).
values at initial and final infinitesimal stages. The constant environment temperature is the parameter designated by T e . On the other hand, subscripts pertain to state coordinates. The popular ‘engine convention’ us used: work generated in an engine, W , is positive, and work generated in a heat pump is negative; this means that a positive work (−W ) is consumed in the heat pump. The sign of the optimal work function V N = max W N defines the working mode for an optimal sequential process as a whole. In engine modes W > 0 and V > 0. In heat-pump modes, W < 0 and V < 0, so working with a function R N = −V N = min(−W N ) is more convenient. Of special attention are two processes: the one which starts with the state T 0 = T e and terminates at an arbitrary T N = T and the one which starts at an arbitrary T 0 = T and terminates at T e . The functions V N are then generalizations of the classical exergy for discrete processes with finite durations.
2. MODELING AND OPTIMIZATION OF OPERATIONS WITH HEAT TRANSFER In a single-stage process (figure 1) the fluid’s temperature changes between T 0 and T 1 , the related heat flux
168
Te un θ n T n + un
(2)
where un = −q n /g n and θ n = τ n − τ n−1 is the free increment of nondimensional time τ at stage n. The time itself is defined by equation (5) below. The specific work (2) has to be minimized subject to the difference constraints T n − T n−1 = un , τ n − τ n−1
τ n − τ n−1 =1 θn
(3)
Equation (2) describes the work supplied to the process in which the controlled fluid is sequentially heated in condensers of N endoreversible heat pumps. Yet, this formulation is valid for both process modes. Writing equation (2) in an equivalent form WN = −
N Te c 1 − n T n T n=1
−
N
cT
e
n=1
un un θ n T n (T n + un )
(2 )
we find that in the limiting case of the process with an infinite number of stages a work integral is obtained in the form of equation (4). The integral applies the equality u = dT /dτ which is valid for the temperature representation of the driving heat per unit overall conductance, Te dT c 1− T Ti Tf (T˙ )2 −Te c dτ T (T + T˙ ) Ti
W ≡ P /Gf = −
Tf
(4)
S. Sieniutycz, Z. Szwast / Exergy Int. J. 1(3) (2001) 166–172
Equation (4) refers to a continuous process in which the fluid (heated in an infinite sequence of inifinitesimal heat pumps or cooled in an infinite sequence of inifinitesimal engines) changes its states between the initial temperature T i and the final temperature T f . The first term is the reversible thermodynamic work W rev (the classical exergy change). The second term is the negative of the ambient temperature and the entropy production. The temperature derivative is with respect to the nondimensional time or the number of heat transfer units. The latter can be linked with the length coordinate, x, or the fluid’s residence time t by an equation τ≡
α av t α av F x= t= Gc ρc χ
(5)
where α is an overall heat transfer coefficient, av —the specific area, F —crossectional area and χ = ρc/(α av ) is a time constant. Note that we are still residing in the realm of endoreversible processes, although now they refer to an infinite number of infinitesimal NCCA stages. Equation (4) proves that it is the entropy production which causes the nonpotential component of the work integral. We note that minimizing the entropy production in a fixed-end sequential problem assures minimum of the work consumption in the heat-pump mode and maximum of work production in the engine mode. Since the first or potential term is path independent, the (nonpotential) entropy production determines the property of the extremal trajectory. Therefore, as implied by equation (4), a common differential equation holds for fixed-end extremals of extremum work and minimum entropy production [6]. For all modes we find T T¨ − T˙ 2 = 0
(6)
Equation (6) is satisfied by the function T (τ ) which solves a simple differential equation, dT /dτ = ξ T
(7)
where the constant ξ is the rate indicator which is positive for the fluid’s heating and negative for fluid’s cooling. An unconstrained extremal is an exponential curve. Consider now extremals of the corresponding multistage process, equations (2) and (3). Since the discrete model is linear with respect to the time interval θ n , a discrete algorithm with a constant Hamiltonian governs the optimal multistage process [7, 8]. The optimal discrete dynamics has the form T n − T n−1 = ξT n θn
(8)
which is a discrete analog of equation (7). The optimal solution asserts that θ n = θ n−1 and n 2 = T n−1 T n+1 (9) T This means that the temperatures T n between the stages n and n + 1 are geometric means of the boundary temperatures. The discrete solution converges to the exponential solution of equation (7) in the limit of an infinite N . In fact, equation (9) is a discrete generalization of the optimality condition known for optimal trajectories of heat exchangers and simulated annealing [9–11]. For the exergy boundary conditions, the optimal work related to equation (9) is a discrete generalization of the continuous finite-time exergy [6].
3. ANALOGIES BETWEEN THERMAL OPERATIONS WITH AND WITHOUT WORK With the substitution u = T − T , which defines a new control variable T , the so-called driving temperature, the minimum work in the heat-pump mode can be described by the optimal performance function R T i , T f , τ f − τ i ≡ min(−P /Gf ) τf Te = min c 1 − T − T dτ i T τ f i e f = h − h − T s − si Tf (T − T )2 + T e min dτ c T T Ti (10) The name ‘driving temperature’ is substantiated by the basic property of T : it is the temperature of an external medium whose controlling effect on the fluid’s temperature T is the same as that in the traditional exchanger of heat or mass (an exchanger without any work production or consumption). It follows from equation (10) that in terms of T the entropy production acquires its classical form in which the heat flux is multiplied by the difference in temperature reciprocals, as known from the theory of traditional heat exchangers. This means, of course, the simplicity of modeling of work-assisted processes in terms of T , and this is the main reason why to introduce T . Likewise, the maximum work in the engine mode is described by the optimal function V = −R, where T and T are linked by the constraint dT /dτ = T − T
(11)
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For multistage processes with heat pumps or engines an analogous discrete picture exists with sums replacing integrals and differential ratios instead of derivatives. The discrete counterpart of the optimal cost function (10) is then the minimum of expression (2).
mass balances
Let us observe that in terms of T the analytical forms of expressions for the entropy production and for the associated constraint (e.g., equation (11)) are precisely those which describe a number of purely dissipative processes i.e. those without work production or consumption. For example, with the nondimensional time τ defined as τ = Gg cg /(Gs cs ), equation (11) describes the temperature change of the solid as the controlled phase in a process in which the gas crosses vertically the bed of the granular solid in a horizontal fluidized exchanger (HFE) provided that the equilibrium between the outlet gas and the outlet solid is attained. Moreover, the integral in the third line of equation (10) with c = cs describes the associated entropy production per unit mass of the controlled solid. Indeed, for the workless HFE process we find T f 1 1 − dQs Sσ = T T Ti Tf 1 1 − T − T dGg /Gs = cg T T Ti Tf (T − T )2 dτ (12) = cs T T Ti
where the state of the controlling phase, enthalpy i and humidity X is that of the bulk of gas, i.e., i = ig and X = Xg . We aim to derive suitable formulae for entropy production which will be useful for evaluation of losses of work potential in thermal machines. The specific entropy produced per unit mass of solid is the path integral over the scalar product of vectors (dI, dΩ) and (1/T − 1/T , µ/T − µ /T ). The difference of Planck potentials governs the mass transfer. Thus, with energy and mass balances (13) T f 1 1 µ µ − dI − − dΩ Sσ = T T T T Ti T f 1 1 − i − is = i T T T µ µ − X − Xs dτ + (14) T T
Thus the purely dissipative process of fluidized heat exchange can model the more difficult process with the thermal machine. The same conclusion holds for cascades with finite number of stages. Such analogies are formal but, nonetheless, they help significantly to model and optimize work-assisted operations in the realm of the entropy production expressions, not work expressions (work terms are absent in equations of purely dissipative systems).
4. GENERALIZED ANALOGIES FOR NONISOTHERMAL MASS TRANSFER Here we shall show that operations in HFE dryers can model work-assisted operations with heat and mass transfer. For drying processes the controlled phase is solid, and the space of solid enthalpy I and solid moisture content Ω is the state space. This choice of variables assures the simplest form of the energy and
170
Gs dI = dGg i − is (I, Ω) Gs dW = dGg X − Xs (I, Ω)
(13)
where the differential of variable τ satisfies dτ = dGg /Gs . With the Hessian of controlling phase entropy s , or the matrix of the second order derivatives of s with respect to i and X , the Taylor expansion of thermodynamic forces in equation (14) yields a positive integral
T f (i − is ) Γ11 Γ12 Sσ = (X − Xs ) Γ21 Γ22 Ti dτ (15) × i − is X − Xs where the positive matrix Γik = −δ 2 s /δx i δx k is the negative Hessian of entropy s . The benefit is the entropy production in an explicit form that is usable for thermal machines. The quantity Sσ for a related multistage process is the corresponding sum over the stages. Equations (15) and its discrete analog represent the continuous and discrete generalizations of equation (12) for the mass transfer coupled with the transer of heat. Optimization of criteria of this sort leads quite generally to a constant intensity of the entropy production along an optimal path [11], in other words in the strictly linear case they satisfy ‘equipartition of the entropy production’ [12]. However, the ‘equipartition principle’ is not really a principle as it is valid only in the case when no constraints are imposed on parameters of the controlling phase (gas). Postquadratic terms in the criterion and nonlinearities in
S. Sieniutycz, Z. Szwast / Exergy Int. J. 1(3) (2001) 166–172
kinetic equations may also cause the violation of the principle. Expression such as equation (15) appears also in the realm of thermal machines, as a representation of their lost work divided by T e . The optimization of equation (15) automatically eliminates the controlling (primed) parameters from Sσ , thus generating the potential Rσ which depends only on the initial and final states of the considered phase and the process duration. However, up to now a precise set of conditions under which equation (15) could serve as a suitable model for a work-assisted system is unknown. Extended studies in finite-time thermodynamics of complex engines and heat pumps with heat and mass transfer are further necessary [13]. In such systems humid gases and hygroscopic solids are utilized by endoreversible heat pumps while exchanging mass and heat. Minimization of the total work consumed over a finite duration leads to a finite-time exergies of gas and solid, Ag and As . Formulae for such exergies follow from the optimal work expressions, equations (2) and (10), when the final states are identified with the states of equilibrium with the environment [13]. With the knowledge of the classical exergy, Arev , a numerical procedure can generate data for both A and min Sσ . Enhanced bounds follow from the finite-time exergy on the work production and consumption. Applications of the theory to multistage drying processes which use heat pumps are described in [14].
5. EXAMPLE We solve here the following example: determine the critically short residence time of the fluid in a pipeline below of which no work production is possible by the system composed of the pipeline and the environment. To solve the problem we search for the particular value of time t at which the upper bound for work W vanishes. Integrating equation (4) along the path (7) subject to the condition W = 0 and for initial time t i = 0 yields in terms of the classical thermal exergy Ex the following relation χ[ln(T /T e )]2 =0 Ex T , T e − cT e tc − χ ln(T /T e )
(16)
from which the critical time tc easily follows as χ[ln(T /T e )]2 tc = χ ln T /T e + Ex (T , T e )/(cT e )
(17)
As the time constant χ = ρc/(α av ) decreases with the heat transfer coefficient α , the critical time tc is
shorter for media with large thermal conductivities. This also means that for media conducting heat perfectly (‘heat superconductors’) the critical time equals zero, i.e., the work production is always possible. On the other hand, for thermally resistive media (i.e., those with small thermal conductivities) critical times tc are always greater than zero, meaning that no mechanical energy production is possible when a medium’s residence time is less than tc (i.e., the system’s length is short enough) whenever temperatures T and T e are fixed.
6. CONCLUSIONS Here we discuss the main results of the paper. We have compared models of work-assisted operations with those describing traditional operations (without work), and we have shown through examples that when the driving temperature T is used as a process variable both operations are described by formally identical mathematical models. This result facilitates modeling of work-assisted operations because the experience gained during modeling of traditional heat and mass exchangers can be transferred to much more difficult systems which produce or consume work. In all these systems, the constraint on process duration forces an engineer to work with finite rates in order to assure required changes in end states of key or controlled media in a finite time. However the same time constraint causes enhanced bounds on the work production and consumption. The upper bound for the work production in any irreversible engine-mode operation is lower than that for the same operation conducted reversibly (between the same boundary thermodynamic states). This means that any real operation working in the engine mode can only produce less mechanical energy than the reversible operation. For fluid’s residence times shorter than the critical time evaluated from equation (17) no work production is possible. Likewise, the lower bound for the work consumption in any irreversible heat-pump-mode operation is higher than that for the same operation conducted reversibly. This means that any real operation working in the heat pump mode can only consume more mechanical energy than the reversible operation. Our quantitative results (equation (2) for multistage operation and equation (4) for continuous one), make it possible to calculate how less work can be produced by engines or how more work must be consumed by heat pumps working in sequential arrangements, for prescribed residence times of controlled fluids. Moreover, equation (17) answers the question whether the work production is at all possible in a given system.
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Acknowledgements
This work was supported by Polish Committee of National Research (Grant 3-T09C-063).
REFERENCES [1] Berry R.S., Kazakov V.A., Sieniutycz S., Szwast Z., Tsirlin A.M., Thermodynamic Optimization of Finite Time Processes, Wiley, Chichester, 2000. [2] Bejan A., Entropy Generation Minimization: the Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes, CRC Press, Boca Raton, 1995. [3] Andresen B., Finite Time Thermodynamics, Univ. of Copenagen, Copenagen, 1983. [4] Sieniutycz S., Endoreversible modeling and optimization of thermal machines by dynamic programming, Chapter 11, in: Wu Ch. (Ed.), Advance in Recent Finite Time Thermodynamics, Nova Science, New York, 1999. [5] Sieniutycz S., Szwast Z., Optimization of multistage thermal machines by a Pontryagin’s-like discrete maximum principle, Chapter 12, in: Wu Ch. (Ed.), Advance in Recent Finite Time Thermodynamics, Nova Science, New York, 1999.
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[6] Sieniutycz S., Hamilton–Jacobi–Bellman theory of dissipative thermal availability, Phys. Rev. 56 (1997) 5051– 5064. [7] Sieniutycz S., Szwast Z., A discrete algorithm for optimization with a constant hamiltonian and its application to chemical engineering, Internat. J. Chem. Engng. 23 (1983) 155–166. [8] Sieniutycz S., Szwast Z., Practice in Optimization, WNT, Warsaw, 1982. [9] Salamon P., Nitzan A., Andresen B., Berry R.S., Minimum entropy production and the optimization of heat engines, Phys. Rev. A 21 (1980) 2115–2129. [10] Andresen B., Gordon J., Constant thermodynamic speed for minimizing eqntropy production in thermodynamic processes and simulated annealing, Phys. Rev. E 50 (1994) 4346–4351. [11] Spirkl W., Ries H., Optimal finite-time endoreversible processes, Phys. Rev. E 52 (1995) 3485–3489. [12] Tondeur D., Kvaalen E., Equipartition of entropy production: an optimality criterion for transfer and separation processes, Ind. Engrg. Chem. Res. 26 (1987) 50–56. [13] Sieniutycz S., Optimal control framework for multistage engines with heat and mass transfer, J. NonEquilibrium Thermodyn. 24 (1999) 40–74. [14] Sieniutycz S., Thermodynamic optimization or work-assisted heating and drying operations, Energy Conversion and Management 41 (18) (2000) 2009–2031.
Thermodynamic framework for work-assisted unit operations Stanislaw Sieniutycz ∗ , Zbigniew Szwast Faculty of Chemical Engineering, Warsaw University of Technology, 1 Warynskiego Street, 00-645 Warsaw, Poland
(Received 16 July 2000, accepted 6 February 2001)
Abstract — We construct thermodynamic criteria for optimization of sequential work-assisted unit operations, especially for heating, evaporation and drying. Constraints take into account dynamics of heat and mass transport and rate of real work consumption. Finiterate, endoreversible models are used. By definition, they refer to systems which have separated in space their internal reversible parts, such as Carnot engines or heat pumps, from their (external) purely irreversible parts, such as thermal resistances. Consequently these models can quantify irreducible losses of classical exergy caused by the resistances. Extremum performance functions for optimal work, which incorporate residual minimum entropy production, are determined in terms of end states, duration and (in discrete processes) number of stages. Analogies between entropy production expressions for work-assisted operations and those without work help to formulate optimization models of the former. Our theory also answers the question whether work production is at all possible in a given system. 2001 Éditions scientifiques et médicales Elsevier SAS
Nomenclature A bg , bs
available energy (exergy) . . . specific exergy of gas and gas in equilibrium with solid . . . . . b specific exergy of controlling phase . . . . . . . . . . . . . . c specific heat at the constant pressure . . . . . . . . . . . . . G mass flux, total flow rate . . . partial and overall conductances g1 , g HTU height of transfer uni . . . . . solid enthalpy at stage n . . . . In i specific enthalpy of controlling phase . . . . . . . . . . . . . . ig , is specific enthalpy of gas and gas at equilibrium with solid . . . N total number of stages in a multistage process n current stage number of a multistage process P n , pn cumulative power output and power output at nth stage . . .
q1
kJ
kJ·kg−1 kJ·kg−1 kJ·K−1 ·kg−1 kg·s kJ·K−1 ·s−1 m kJ·kg−1 kJ·kg−1 kJ·kg−1
kJ·s−1
∗ Correspondence and reprints.
E-mail address: [email protected] (S. Sieniutycz).
166
driving heat in the engine mode of the stage . . . . . . . . . . . kJ·s−1 n R (x, t) optimal work function of cost type in terms of state and time kJ·kg−1 S solid entropy, entropy of controlled phase . . . . . . . . . . kJ·K−1 s specific entropy of controlling phase, (gas) . . . . . . . . . . . kJ·K−1 ·kg−1 Sσ specific entropy production . . kJ·kg−1 ·K−1 ·s−1 T temperature of controlled phase (solid) . . . . . . . . . . . . . . K Te constant equilibrium temperature of reservoir . . . . . . . . . . . K T temperature of controlling phase (gas) . . . . . . . . . . . . . . K t physical time, contact time . . s un = T n /τ n rate of the temperature change as the control variable . . . . . . . . K V ≡ max W optimal work function of profit type . . . . . . . . . . kJ·K−1 W ≡ P /G total specific work or total power per unit mass flux . kJ·K−1 n W moisture content in solid from stage n . . . . . . . . . . . . . kg−1 ·kg−1 X absolute humidity of controlling phase (gas) . . . . . . . . . . . kg−1 ·kg−1 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1164-0235(01)00022-X/FLA
S. Sieniutycz, Z. Szwast / Exergy Int. J. 1(3) (2001) 166–172
x α η θn
transfer area coordinate . . . . overall heat transfer coefficient = p/q1 first-law efficiency free interval of an independent variable or nondimensional time interval at stage n chemical potential of kth component . . . . . . . . . . . coefficient of outlet gas utilization nondimensional time, number of the heat transfer units (x/HTU )
µk µ τ
m kJ·m−2 ·K−1 ·s−1
kJ·kg−1
Subscripts g i s 1, 2
gas ith state variable saturation, solid first and second fluid
Superscrits e f i k
environment, equilibrium final state initial state or n number of kth or nth stage
1. INTRODUCTION This work analyzes thermodynamic criteria for optimization of work-assisted operations and compares these criteria with those for conventional operations (without work). Basically we display here a quantitative analysis for power criteria associated with production or consumption of mechanical work and for the criterion of entropy production; it is one of the main objectives of this work to show how all these criteria can be reconciled in the realm of thermal machines and traditional exchangers (without work). Our work contributes to the quantitative theory of work production and consumption in single-stage and multi-stage thermal machines in which the effect of fluxes on efficiencies is essential. For sequential operations with thermal machines, such as multistage heat pumps, total power input is minimized at constraints which describe dynamics of energy and mass exchange. The results are optimal work functions in terms of end states, duration and (in discrete processes) number of stages. Modeling a general work-assisted operation is a difficult task. However, formal analogies do exist between entropy production expressions in workassisted and in conventional operations. These analogies are helpful to develop suitable criteria and models. We shall also answer the question whether work production is at all possible in a given system. This problem is not
completely solved in the contemporary theory of thermal machines based on the finite-time thermodynamics (FTT; Refs. [1–3], to which the reader is referred for an extensive literature review of earlier issues). To begin with we consider the heat transfer-driven work generation (consumption) in an ‘endoreversible’ thermal machine, an engine (figure 1) or heat pump, which interacts with a high-T fluid (e.g., drying gas) flowing with the mass flux Gf [4, 5]. The multistage production (consumption) of work requires to use the sequence of Novikov–Chambadal–Curzon–Ahlborn (NCCA) stages (figure 2). In an endoreversible engine a resource fluid drives the Carnot engine from which the work is taken out, in an endoreversible heat pump a fluid (e.g., drying agent) is driven in the condenser of the Carnot heat pump to which work is supplied; in both cases the second fluid is an infinite reservoir. The fluids are of finite thermal conductivity, hence there are finite thermal resistances in the system. In a multistage heating operation the fluid’s temperature increases at each stage; the whole operation is described by the sequence T 0 , T 1 , . . . , T N . Superscripts refer to the stage number; this rule includes also constant process parameters and initial and final values of state variables of a limiting continuous process as they are respectively inlet and outlet
Figure 1. A single-stage NCCA engine as an operation with active heat exchange between two fluids (in the engine mode T1 0).
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is q1 = −Gf c(T 1 − T 0 ), where Gf is the fluid flow and c-fluid’s specific heat. The specific work produced in a single endoreversible engine or that consumed in a single endoreversible heat pump is [4, 5] q1 Te (1) W ≡ −p/Gf = 1 − T − q1 /g Gf where the bracketed expression is the first-law efficiency. Here p is the power output, g—an overall thermal conductance of thermal machine related to an overall heat transfer coefficient, α . In multistage systems (figure 2) one should sum expressions such as equation (1) over stages. Casting the problem in the format of a discrete maximum principle we arrive at the discrete functional of consumed work N −W N = c 1− n=1
Figure 2. A multi-stage NCCA engine as a sequential operation with many Carnot loops to accomplish active heat exchange between two fluids (in the engine mode T1 0).
values at initial and final infinitesimal stages. The constant environment temperature is the parameter designated by T e . On the other hand, subscripts pertain to state coordinates. The popular ‘engine convention’ us used: work generated in an engine, W , is positive, and work generated in a heat pump is negative; this means that a positive work (−W ) is consumed in the heat pump. The sign of the optimal work function V N = max W N defines the working mode for an optimal sequential process as a whole. In engine modes W > 0 and V > 0. In heat-pump modes, W < 0 and V < 0, so working with a function R N = −V N = min(−W N ) is more convenient. Of special attention are two processes: the one which starts with the state T 0 = T e and terminates at an arbitrary T N = T and the one which starts at an arbitrary T 0 = T and terminates at T e . The functions V N are then generalizations of the classical exergy for discrete processes with finite durations.
2. MODELING AND OPTIMIZATION OF OPERATIONS WITH HEAT TRANSFER In a single-stage process (figure 1) the fluid’s temperature changes between T 0 and T 1 , the related heat flux
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Te un θ n T n + un
(2)
where un = −q n /g n and θ n = τ n − τ n−1 is the free increment of nondimensional time τ at stage n. The time itself is defined by equation (5) below. The specific work (2) has to be minimized subject to the difference constraints T n − T n−1 = un , τ n − τ n−1
τ n − τ n−1 =1 θn
(3)
Equation (2) describes the work supplied to the process in which the controlled fluid is sequentially heated in condensers of N endoreversible heat pumps. Yet, this formulation is valid for both process modes. Writing equation (2) in an equivalent form WN = −
N Te c 1 − n T n T n=1
−
N
cT
e
n=1
un un θ n T n (T n + un )
(2 )
we find that in the limiting case of the process with an infinite number of stages a work integral is obtained in the form of equation (4). The integral applies the equality u = dT /dτ which is valid for the temperature representation of the driving heat per unit overall conductance, Te dT c 1− T Ti Tf (T˙ )2 −Te c dτ T (T + T˙ ) Ti
W ≡ P /Gf = −
Tf
(4)
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Equation (4) refers to a continuous process in which the fluid (heated in an infinite sequence of inifinitesimal heat pumps or cooled in an infinite sequence of inifinitesimal engines) changes its states between the initial temperature T i and the final temperature T f . The first term is the reversible thermodynamic work W rev (the classical exergy change). The second term is the negative of the ambient temperature and the entropy production. The temperature derivative is with respect to the nondimensional time or the number of heat transfer units. The latter can be linked with the length coordinate, x, or the fluid’s residence time t by an equation τ≡
α av t α av F x= t= Gc ρc χ
(5)
where α is an overall heat transfer coefficient, av —the specific area, F —crossectional area and χ = ρc/(α av ) is a time constant. Note that we are still residing in the realm of endoreversible processes, although now they refer to an infinite number of infinitesimal NCCA stages. Equation (4) proves that it is the entropy production which causes the nonpotential component of the work integral. We note that minimizing the entropy production in a fixed-end sequential problem assures minimum of the work consumption in the heat-pump mode and maximum of work production in the engine mode. Since the first or potential term is path independent, the (nonpotential) entropy production determines the property of the extremal trajectory. Therefore, as implied by equation (4), a common differential equation holds for fixed-end extremals of extremum work and minimum entropy production [6]. For all modes we find T T¨ − T˙ 2 = 0
(6)
Equation (6) is satisfied by the function T (τ ) which solves a simple differential equation, dT /dτ = ξ T
(7)
where the constant ξ is the rate indicator which is positive for the fluid’s heating and negative for fluid’s cooling. An unconstrained extremal is an exponential curve. Consider now extremals of the corresponding multistage process, equations (2) and (3). Since the discrete model is linear with respect to the time interval θ n , a discrete algorithm with a constant Hamiltonian governs the optimal multistage process [7, 8]. The optimal discrete dynamics has the form T n − T n−1 = ξT n θn
(8)
which is a discrete analog of equation (7). The optimal solution asserts that θ n = θ n−1 and n 2 = T n−1 T n+1 (9) T This means that the temperatures T n between the stages n and n + 1 are geometric means of the boundary temperatures. The discrete solution converges to the exponential solution of equation (7) in the limit of an infinite N . In fact, equation (9) is a discrete generalization of the optimality condition known for optimal trajectories of heat exchangers and simulated annealing [9–11]. For the exergy boundary conditions, the optimal work related to equation (9) is a discrete generalization of the continuous finite-time exergy [6].
3. ANALOGIES BETWEEN THERMAL OPERATIONS WITH AND WITHOUT WORK With the substitution u = T − T , which defines a new control variable T , the so-called driving temperature, the minimum work in the heat-pump mode can be described by the optimal performance function R T i , T f , τ f − τ i ≡ min(−P /Gf ) τf Te = min c 1 − T − T dτ i T τ f i e f = h − h − T s − si Tf (T − T )2 + T e min dτ c T T Ti (10) The name ‘driving temperature’ is substantiated by the basic property of T : it is the temperature of an external medium whose controlling effect on the fluid’s temperature T is the same as that in the traditional exchanger of heat or mass (an exchanger without any work production or consumption). It follows from equation (10) that in terms of T the entropy production acquires its classical form in which the heat flux is multiplied by the difference in temperature reciprocals, as known from the theory of traditional heat exchangers. This means, of course, the simplicity of modeling of work-assisted processes in terms of T , and this is the main reason why to introduce T . Likewise, the maximum work in the engine mode is described by the optimal function V = −R, where T and T are linked by the constraint dT /dτ = T − T
(11)
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For multistage processes with heat pumps or engines an analogous discrete picture exists with sums replacing integrals and differential ratios instead of derivatives. The discrete counterpart of the optimal cost function (10) is then the minimum of expression (2).
mass balances
Let us observe that in terms of T the analytical forms of expressions for the entropy production and for the associated constraint (e.g., equation (11)) are precisely those which describe a number of purely dissipative processes i.e. those without work production or consumption. For example, with the nondimensional time τ defined as τ = Gg cg /(Gs cs ), equation (11) describes the temperature change of the solid as the controlled phase in a process in which the gas crosses vertically the bed of the granular solid in a horizontal fluidized exchanger (HFE) provided that the equilibrium between the outlet gas and the outlet solid is attained. Moreover, the integral in the third line of equation (10) with c = cs describes the associated entropy production per unit mass of the controlled solid. Indeed, for the workless HFE process we find T f 1 1 − dQs Sσ = T T Ti Tf 1 1 − T − T dGg /Gs = cg T T Ti Tf (T − T )2 dτ (12) = cs T T Ti
where the state of the controlling phase, enthalpy i and humidity X is that of the bulk of gas, i.e., i = ig and X = Xg . We aim to derive suitable formulae for entropy production which will be useful for evaluation of losses of work potential in thermal machines. The specific entropy produced per unit mass of solid is the path integral over the scalar product of vectors (dI, dΩ) and (1/T − 1/T , µ/T − µ /T ). The difference of Planck potentials governs the mass transfer. Thus, with energy and mass balances (13) T f 1 1 µ µ − dI − − dΩ Sσ = T T T T Ti T f 1 1 − i − is = i T T T µ µ − X − Xs dτ + (14) T T
Thus the purely dissipative process of fluidized heat exchange can model the more difficult process with the thermal machine. The same conclusion holds for cascades with finite number of stages. Such analogies are formal but, nonetheless, they help significantly to model and optimize work-assisted operations in the realm of the entropy production expressions, not work expressions (work terms are absent in equations of purely dissipative systems).
4. GENERALIZED ANALOGIES FOR NONISOTHERMAL MASS TRANSFER Here we shall show that operations in HFE dryers can model work-assisted operations with heat and mass transfer. For drying processes the controlled phase is solid, and the space of solid enthalpy I and solid moisture content Ω is the state space. This choice of variables assures the simplest form of the energy and
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Gs dI = dGg i − is (I, Ω) Gs dW = dGg X − Xs (I, Ω)
(13)
where the differential of variable τ satisfies dτ = dGg /Gs . With the Hessian of controlling phase entropy s , or the matrix of the second order derivatives of s with respect to i and X , the Taylor expansion of thermodynamic forces in equation (14) yields a positive integral
T f (i − is ) Γ11 Γ12 Sσ = (X − Xs ) Γ21 Γ22 Ti dτ (15) × i − is X − Xs where the positive matrix Γik = −δ 2 s /δx i δx k is the negative Hessian of entropy s . The benefit is the entropy production in an explicit form that is usable for thermal machines. The quantity Sσ for a related multistage process is the corresponding sum over the stages. Equations (15) and its discrete analog represent the continuous and discrete generalizations of equation (12) for the mass transfer coupled with the transer of heat. Optimization of criteria of this sort leads quite generally to a constant intensity of the entropy production along an optimal path [11], in other words in the strictly linear case they satisfy ‘equipartition of the entropy production’ [12]. However, the ‘equipartition principle’ is not really a principle as it is valid only in the case when no constraints are imposed on parameters of the controlling phase (gas). Postquadratic terms in the criterion and nonlinearities in
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kinetic equations may also cause the violation of the principle. Expression such as equation (15) appears also in the realm of thermal machines, as a representation of their lost work divided by T e . The optimization of equation (15) automatically eliminates the controlling (primed) parameters from Sσ , thus generating the potential Rσ which depends only on the initial and final states of the considered phase and the process duration. However, up to now a precise set of conditions under which equation (15) could serve as a suitable model for a work-assisted system is unknown. Extended studies in finite-time thermodynamics of complex engines and heat pumps with heat and mass transfer are further necessary [13]. In such systems humid gases and hygroscopic solids are utilized by endoreversible heat pumps while exchanging mass and heat. Minimization of the total work consumed over a finite duration leads to a finite-time exergies of gas and solid, Ag and As . Formulae for such exergies follow from the optimal work expressions, equations (2) and (10), when the final states are identified with the states of equilibrium with the environment [13]. With the knowledge of the classical exergy, Arev , a numerical procedure can generate data for both A and min Sσ . Enhanced bounds follow from the finite-time exergy on the work production and consumption. Applications of the theory to multistage drying processes which use heat pumps are described in [14].
5. EXAMPLE We solve here the following example: determine the critically short residence time of the fluid in a pipeline below of which no work production is possible by the system composed of the pipeline and the environment. To solve the problem we search for the particular value of time t at which the upper bound for work W vanishes. Integrating equation (4) along the path (7) subject to the condition W = 0 and for initial time t i = 0 yields in terms of the classical thermal exergy Ex the following relation χ[ln(T /T e )]2 =0 Ex T , T e − cT e tc − χ ln(T /T e )
(16)
from which the critical time tc easily follows as χ[ln(T /T e )]2 tc = χ ln T /T e + Ex (T , T e )/(cT e )
(17)
As the time constant χ = ρc/(α av ) decreases with the heat transfer coefficient α , the critical time tc is
shorter for media with large thermal conductivities. This also means that for media conducting heat perfectly (‘heat superconductors’) the critical time equals zero, i.e., the work production is always possible. On the other hand, for thermally resistive media (i.e., those with small thermal conductivities) critical times tc are always greater than zero, meaning that no mechanical energy production is possible when a medium’s residence time is less than tc (i.e., the system’s length is short enough) whenever temperatures T and T e are fixed.
6. CONCLUSIONS Here we discuss the main results of the paper. We have compared models of work-assisted operations with those describing traditional operations (without work), and we have shown through examples that when the driving temperature T is used as a process variable both operations are described by formally identical mathematical models. This result facilitates modeling of work-assisted operations because the experience gained during modeling of traditional heat and mass exchangers can be transferred to much more difficult systems which produce or consume work. In all these systems, the constraint on process duration forces an engineer to work with finite rates in order to assure required changes in end states of key or controlled media in a finite time. However the same time constraint causes enhanced bounds on the work production and consumption. The upper bound for the work production in any irreversible engine-mode operation is lower than that for the same operation conducted reversibly (between the same boundary thermodynamic states). This means that any real operation working in the engine mode can only produce less mechanical energy than the reversible operation. For fluid’s residence times shorter than the critical time evaluated from equation (17) no work production is possible. Likewise, the lower bound for the work consumption in any irreversible heat-pump-mode operation is higher than that for the same operation conducted reversibly. This means that any real operation working in the heat pump mode can only consume more mechanical energy than the reversible operation. Our quantitative results (equation (2) for multistage operation and equation (4) for continuous one), make it possible to calculate how less work can be produced by engines or how more work must be consumed by heat pumps working in sequential arrangements, for prescribed residence times of controlled fluids. Moreover, equation (17) answers the question whether the work production is at all possible in a given system.
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Acknowledgements
This work was supported by Polish Committee of National Research (Grant 3-T09C-063).
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