Diffuser and Nozzle Design Optimization by Entropy Generation Minimization
Abstract
:1. Introduction
2. Head Loss, Dissipation and Entropy Generation
- A -value, defined for all kinds of conduit components like bends, trijunctions but also straight pipe segments, often is defined in terms of pressure differences that occur along that component. For example a straight pipe of diameter D and length L is characterized by a friction factor f with
- For conduit components with different cross section areas upstream and downstream (like with diffusers and nozzles, c.f. Figure 1) there should be a careful interpretation with respect to pressure differences which are often measured in order to characterize the components. If is measured with and , being the pressure in cross sections ① and ② in Figure 1, there is
- a change of pressure due to the change of cross section, , corresponding to the difference in kinetic energy in both cross sections.
- a loss of total head due to losses in the flow field, . This can formally be subdivided into two parts, i.e., . Here, is the frictional loss in the adjacent tangents (index: AT) from the conduit component to the cross section ① and ②, respectively. It is determined under the assumption of a fully developed and undisturbed flow in the tangents upstream and downstream. Then corresponds to the loss of total head due to (not in) the conduit component (index: CC). It can be uniquely linked to the head loss coefficient . It is the only part inThis is sketched in Figure 2 where X is a parameter of the geometry that can be altered during the optimization process. With the definition of given above this part of the measured (corresponding to the head loss coefficient) is independent of the exact location of ① and ② in Figure 1, provided both cross sections are outside the zones of influence upstream and downstream of the component.In Figure 2b it has been anticipated that in a nozzle can be negative, i.e., the nozzle may reduce the head loss in the adjacent tangents compared to the assumed loss due to a fully developed and undisturbed flow.
- Losses from a thermodynamic point of view are best described as losses of exergy, sometimes called losses of available work (exergy: maximum theoretical work obtainable from the energy interacting with the environment to equilibrium). In a power cycle, for example, this is especially important, since lost exergy is no longer available in the turbine of that cycle resulting in a reduced cycle efficiency, see [2] for a comprehensive discussion of this issue.Dissipation of mechanical energy, which is directly linked to the head loss coefficient in (2), isSince according to (8) it depends on the temperature level how much exergy is lost with a certain dissipation the head loss coefficient defined in (1) is not a general measure for the exergy lost in a conduit component.Hence, in addition to a second coefficient called exergy loss coefficient should be introduced. With defined as
- During the optimization process described hereafter, -values of the diffusers and nozzles will have to be determined. According to (2) this can either be done by the determination of φ (dissipation rate per mass flux) or of s (entropy generation rate per mass flux). We definitely prefer s since it is the more fundamental quantity of the conversion process (mechanical → thermal energy). A further argument in favor of the specific entropy is that then a common quantity exists when also losses (of exergy) in a superimposed heat transfer process are accounted for, see [1] for more details of this convective heat transfer situation.
3. Review of Literature
3.1. Literature about Entropy in General
3.2. Literature about Entropy Generation
3.3. Literature about Optimization Processes
4. Entropy Generation as an Optimization Criterion
4.1. Determination of the Overall Entropy Generation Rate
4.2. An Example
- the additional entropy generation upstream,
- the entropy generation in the component (nozzle),
- the additional entropy generation downstream, , which here is negative
K | |||
---|---|---|---|
(24) | 1.088 | 1.069 | −0.125 |
(24) and | 1.077 | 1.060 | 0.018 |
1.0 | 1.0 | 0.897 |
5. Nozzle Optimization
5.1. Numerical Details
5.2. Nozzle Optimization for the Polynomial Wall Shape,
6. Diffuser Optimization
6.1. Diffuser Optimization for the Polynomial Wall Shape;
6.2. Diffuser Optimization for a Non-Straight Wall Shape;
7. Conclusions
Acknowledgements
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Schmandt, B.; Herwig, H. Diffuser and Nozzle Design Optimization by Entropy Generation Minimization. Entropy 2011, 13, 1380-1402. https://doi.org/10.3390/e13071380
Schmandt B, Herwig H. Diffuser and Nozzle Design Optimization by Entropy Generation Minimization. Entropy. 2011; 13(7):1380-1402. https://doi.org/10.3390/e13071380
Chicago/Turabian StyleSchmandt, Bastian, and Heinz Herwig. 2011. "Diffuser and Nozzle Design Optimization by Entropy Generation Minimization" Entropy 13, no. 7: 1380-1402. https://doi.org/10.3390/e13071380