Potential and Practicality of Finite Time Thermodynamics a report done for

by Robby Whitesell

February 15, 2007

In partial fulfillment of the requirements of the course ME 620 Advanced Engineering Thermodynamics D. P. Sekulic

DEPARTMENT OF MECHANICAL ENGINEERING UNIVERSITY OF KENTUCKY

Explore in detail advanced thermodynamic analysis/theory/approach and establish your own position regarding its meaning, correctness, usefulness and relevance for engineering study and/or scientific inquiry. Abstract This paper looks both sides of the argument on the practicality and correctness of a relatively new branch of thermodynamics, Finite Time Thermodynamics. This paper is meant to highlight some key points on both sides and form and opinion based on these arguments. While Finite Time Thermodynamics might not be theoretically a sound concept, it does have uses if utilized correctly. These uses far outweigh the theoretical discrepancies. Introduction The ideas of Finite Time Thermodynamics (FTT) have only been around for a few decades yet there is much debate on this topic. There are two distinct factions (as expected), those who support FTT for its practicality and those who oppose it because of its theoretical incorrectness.

This paper broadly looks at and discusses FTT while

conveying opinions on its theoretical merit and practical usefulness as a real world design tool and goal. Throughout this paper, FTT will be discussed through its model of a power plant. This model consists of an endoreversible heat engine and two irreversible heat exchangers that act as heat reservoirs.4 It is important to clarify what exactly is being discussed because FTT, in its most general form, is only a field or view that attempts to bridge the established fields of classical thermodynamics and heat transfer.3 It attempts to model reversible processes which normally require infinite time to complete or infinitely large heat transfer areas in heat exchangers as finite time or size processes.1 FTT tries to give designers and engineers a more realistic goal for efficiency of a heat engine since the ideal Carnot efficiency is beyond real world reach.1 The Carnot efficiency of a heat engine is well known and describes the maximum efficiency for a heat engine. However, it assumes that all processes in the heat engine are

reversible.1 Clearly, in real world applications this is not the case. Thus, FTT attempts to introduce a new efficiency that represents a more realistic goal for a heat engine. The typical FTT model of a power plant is shown below and used as a basis for the following analysis.

T τ+ T+

2

Q+

3 W

Endoreversible Chamber Tτ-

Q-

1 S1

4 S2

S

T-S diagram of FTT model of a power plant operation between two heat reservoirs.3

Here you can see that the engine still operates between two reservoirs, but each one is limited to a constant temperature. Heat is added and taken from the engine through irreversible temperature differences between the reservoir temperature and the engine temperature. This heat flux is used to create work out of the engine. Curzon-Ahlborn Efficiency Derivation Below is a derivation of the Curzon-Ahlborn efficiency based on the FTT model of a power plant. The presentation here is merely meant as a guide and only highlights the key steps. Most of the algebra is left out for sake of space. τ+ and τ- are given.12 First, start with the formal definition of efficiency. .

η=

W .

Q+

(1)

Q- and Q+ represent the heat transfer rates between the heat sink and heat source, respectively. .

Q− = c− (T− − τ − ) .

Q + = c+ (τ + − T+ )

(2) (3)

The formal definition of Work as it applies to this system .

.

.

W = Q+ − Q−

(4)

By combining (1), (2) and (3), we get the following: .

η = 1−

Q− .

Q+

= 1−

T− T+

(5)

By looking at the term with temperature in it from (5) and subbing back in (2) and (3), we can find a relation for T-.

T− c− (T− − τ − ) c−T+τ − = ⇒ T− = T+ c+ (τ + − T+ ) (c+ + c− )T+ − c+τ +

(6)

With a relation for T- in place, we now need a relation for T+. To find this, we take the derivative of (4) with respect to T+ after subbing (2), (3) and (6) into it and setting it equal to zero. Thus, power output is maximized for the efficiency equation given later.11

T+ =

1 (c+τ + ± c− τ +τ − c+ + c −

(7)

Now, we insert (7) into (6)2 and use that along with (7) to get the following: T− (c−τ − ± c+ τ +τ − ) = T+ (c+τ + ± c− τ +τ − )

(8)

Equation (8) then becomes the Curzon-Ahlborn efficiency associated with an FTT model of a heat engine.

ηCA = 1 −

τ− τ+

(9)

The derivation of this new efficiency is clean and yields a value that will always be lower than the Carnot efficiency for the same heat engine.1 This derivation will work for both steady flow and reciprocating processes.3,9 This has been shown several times as well.

Theoretical Problems Although the derivation of the Curzon-Ahlborn efficiency is clear and neat, there are a few problems hidden just beneath the surface. But, to understand these we need to look at a major component of the standard FTT power plant model, the endoreversible heat engine. Endoreversibility means that inside the engine, all processes can be considered reversible and the only irreversibilities are outside.4,5,8 This model is acceptable and makes sense on its own4, but in the case of the FTT power plant model, it does not. In the FTT power plant the endoreversible chamber is connected to a heat source and heat sink.4 This connection logically and practically is accomplished by heat exchangers.4,6 These heat exchangers are not infinitely large thus they have irreversibilities associated with them. Therefore, there is entropy exchange and generation in the heat exchangers.4 These entropy changes and generations are interconnected between the two fluids and thus will enter the endoreversible chamber as heat is exchanged.4 This violates the endoreversible chamber definition. In essence, the FTT power plant model asks the boundary between the endoreversible chamber and the heat exchangers to be reversible when looking from the chamber side and irreversible when looking from the heat exchanger side.4 Another interesting problem arises from one of the main ideas behind FTT. Reversible processes take infinitely long to complete and therefore produce no power. The whole idea of FTT is to get rid of such reversible processes by giving them a finite time to reach completion. The FTT power plant model uses an endoreversible chamber (which uses reversible processes) as the centerpiece where all the action takes place.7 It is very hard

to put faith into a concept that allows violations of its most basic ideas to exist within the defined system. From the above to reasons it is clear that FTT does have some problems associated with it. The equations work out nicely, but the theory behind them is questionable. It is clear that to accomplish a theoretically correct FTT model of anything careful consideration must be taken so that no basic definitions are violated.5,8,10 In the above power plant model, the definition of endoreversibility is violated through entropy exchange. The fact that problems like this can arise so easily is a major problem. Clarification of FTT and Practicality While some of the intricacies behind FTT are not theoretically correct, that does not, to me, mean it should be disregarded as irrelevant. It is important to look closely at the exact goals of FTT and then see if these are met. The very broad goal of FTT is to make a realistic model out of an unrealistic process. Finding a way to model an infinitely long process or infinitely large heat exchanger can be done, but that has no use in the real world. Engineers and designers want guidelines or rules of thumb or generalities to help guide their decision making processes. A way to accomplish this is by using and FTT model of a power plant. Some say that to make a good model you need to include all of the irreversibilities in the power plant.7 However, most models do not include every detail of the actual system. They include what is needed to demonstrate a point. To take full advantage of FTT you need a solid grasp of the problem and system definition. These two things will greatly enhance the capabilities of the model. Also, you need to know which irreversibilities you want to leave out and which to leave in.8,10 Just like any modeling tool, the setup is most important. FTT is meant to general; to provide a cheaper and faster analysis of a problem than running a complete analysis.8 It is also important to realize that the Curzon-Ahlborn efficiency equation derived above is not a limit on efficiency, but rather a guide.2,10 The Carnot efficiency equation is a limit and will be greater than then Curzon-Ahlborn or real world efficiency by definition.2 The Curzon-Ahlborn equation is an efficiency equation of the FTT heat

engine at maximum power.1 Thus, it is possible to run a heat engine at less than full power to attain an efficiency greater than the Curzon-Ahlborn efficiency.5 To arrive at a usable and meaningful solution using an FTT approach it is critical to completely define the system and know exactly was is being tested. The more that is known about the problem, the better the FTT model will be. It is also important to only compare similar quantities. When using the Curzon-Ahlborn efficiency equation is very critical to only apply it to engines operating at maximum power.

Conclusion While FTT might easily violate classic thermodynamic laws, it is irresponsible to completely dismiss its findings and results. Engineers and scientists are always on the lookout for better ways to measure performance limits to help create better designs. Although FTT could violate definitions in some of its uses and is not accepted by some scholars, it is not something to be disregarded. FTT can provide workable performance guidelines if used properly. But, FTT cannot be seen as the one and only guideline. It is merely another way to quickly and cheaply gauge the potential performance of a design to determine if it is worth exploring. The more guidelines, helpful ideas, thoughts and ways of thinking and engineer possesses or has access too, the better his designs and inventions.

To me, the potential benefits of FTT greatly outweigh the problems

associated with some of its more intricate details and happenings.

References 1

Curzon, F. L., Ahlborn, B., AJP Volume 43, January 1975, pp.22-24 De Vos, A., Am. J. Phys. Vol. 53, No. 6, June 1985, pp.570-573 3 Wu, C., Kiang, R. L., Lopardo, V. J. Karpouzian, G. N., Int. J. of Mech. Eng. Edu., Vol 21, No. 4, 1993, pp.337-346 4 Sekulic, D. P., J. Appl. Phys. Vol. 83, No. 9, 1998 pp. 4561-4565 5 Andresen, B., J. Appl. Phys, Vol. 90, No. 12, 2001, pp. 6557-6559 6 Sekulic, D. P., J. Appl. Phys, Vol. 90, No. 12, 2001, pp. 6560-6561 7 Gyftopoulos, E. P., Energy, Vol. 24, 1999, pp 1035-1039 8 Salamon, P., Some Issues in Finite Time Thermodynamics, 2001 9 Andresen, B., Berry, R. S., Ondrechen, M. J., Salamon, P., Acc. Chem. Res., 1984, 17, pp. 266-271 10 Salamon, P., A Contrast Between the Physical and the Engineering Approaches to Finite-Time Thermodynamic Models and Optimizations, 1998 11 Ladino-Luna D., Revista Mexicana de Fisica, Vol. 48 No. 6, 2002, pp. 86-90 12 It is important to note that this derivation has been carried out many, many times and this representation can be found in several forms in almost any of the sources listed here. 2