# Diffuser and Nozzle Design Optimization by Entropy Generation Minimization

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Basic design of diffusers and nozzles within a flow system; ①, ② locations beyond the influence of the conduit component.

## 2. Head Loss, Dissipation and Entropy Generation

- A $\mathbf{K}$-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$$\mathbf{K}=\text{f}\frac{L}{D}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\text{f}\equiv \frac{-\text{d}p/\text{d}x}{\varrho {u}_{m}^{2}/\left(2D\right)}$$$$\mathbf{K}\equiv \frac{\Delta p}{\varrho {u}_{m}^{2}/2}$$
- 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 $\Delta p={p}_{2}-{p}_{1}$ is measured with ${p}_{1}$ and ${p}_{2}$, 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, $\Delta {p}_{CS}^{12}$, corresponding to the difference in kinetic energy in both cross sections.
- a loss of total head due to losses in the flow field, $\Delta {p}_{L}^{12}$. This $\Delta {p}_{L}^{12}$ can formally be subdivided into two parts, i.e., $\Delta {p}_{L}^{12}=\Delta {p}_{L,AT}^{12}+\Delta {p}_{L,CC}^{12}$. Here, $\Delta {p}_{L,AT}^{12}$ 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 $\Delta {p}_{L,CC}^{12}$ 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 $\mathbf{K}$. It is the only part in$$\Delta p\equiv {p}_{2}-{p}_{1}=\Delta {p}_{CS}^{12}-\left[\Delta {p}_{L,AT}^{12}+\Delta {p}_{L,CC}^{12}\right]$$This 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 $\Delta {p}_{L,CC}^{12}$ given above this part of the measured $\Delta p$ (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 $\Delta {p}_{L,CC}^{12}$ 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), is$$\phi ={T}_{m}s$$$$s=\frac{{e}_{L}}{{T}_{0}}$$$$\phi =\frac{{T}_{m}}{{T}_{0}}{e}_{L}$$Since according to (8) it depends on the temperature level ${T}_{m}$ how much exergy is lost with a certain dissipation the head loss coefficient $\mathbf{K}$ defined in (1) is not a general measure for the exergy lost in a conduit component.Hence, in addition to $\mathbf{K}$ a second coefficient called exergy loss coefficient should be introduced. With ${\mathbf{K}}^{\text{E}}$ defined as$${\mathbf{K}}^{\text{E}}\equiv \frac{{e}_{L}}{{u}_{m}^{2}/2}=\frac{{T}_{0}s}{{u}_{m}^{2}/2}$$$$\mathbf{K}=\frac{{T}_{m}}{{T}_{0}}{\mathbf{K}}^{\text{E}}$$
- During the optimization process described hereafter, $\mathbf{K}$-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.

**Figure 2.**Typical overall pressure difference and its interpretation according to (5) for an optimization with respect to an optimization parameter X (not specified here).

## 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

**Figure 3.**Different parts of the flow field determined or influenced by the conduit component. ${V}_{c}$: volume of the component itself; ${V}_{u}$: upstream volume affected by the component; ${V}_{d}$: downstream volume affected by the component.

^{3}mK)) when integrated over a certain volume of the flow field results in the overall entropy generation rate $\dot{S}$ or its specific value $s=\dot{S}/\dot{m}$ in this volume. Outside of the diffuser or nozzle, however, only the additional entropy generation has to be determined, i.e., only that parts of ${\dot{S}}^{\prime \prime \prime}$ that do not exist in the fully developed and undisturbed flow in the adjacent tangents upstream and downstream. In Figure 4 the relevant entropy generation rate $\dot{S}=\phi \dot{m}/{T}_{m}$ with its three parts is shown as the integral over ${\dot{S}}^{\prime \prime \prime}$ and $({\dot{S}}^{\prime \prime \prime}-{\dot{S}}_{0}^{\prime \prime \prime})$, respectively. Away from the conduit component $({\dot{S}}^{\prime \prime \prime}-{\dot{S}}_{0}^{\prime \prime \prime})$ asymptotically tends to zero so that a finite value has to be set when the upstream and downstream lengths of influence should be determined.

**Figure 4.**Determination of the overall entropy generation rate due to a conduit component. $\Delta {\phi}_{u}$: additional specific dissipation upstream of the component; $\phantom{\Delta}{\phi}_{c}$: specific dissipation in the component; $\Delta {\phi}_{d}$: additional specific dissipation downstream of the component.

#### 4.2. An Example

**Figure 6.**Cross sectional entropy generation rate for the nozzle at ${\mathbf{Re}}_{1}=50000$ with $L=2{D}_{1}$ and ${R}_{m}=0.375{D}_{1}$, dark: loss inside the nozzle, light: gain downstream of the nozzle.

- the additional entropy generation upstream, $\Delta {\phi}_{u}\dot{m}/{T}_{m}=\Delta {s}_{u}\dot{m}$
- the entropy generation in the component (nozzle), ${\phi}_{c}\dot{m}/{T}_{m}={s}_{c}\dot{m}$
- the additional entropy generation downstream, $\Delta {\phi}_{d}\dot{m}/{T}_{m}=\Delta {s}_{d}\dot{m}$, which here is negative

_{1}, f

_{2}are the friction factors of the fully developed pipe flows, ${L}_{1}={L}_{u}$ ,${L}_{2}={L}_{d}$ are the upstream and downstream tangent lengths, and $\mathbf{K}$ is the head loss coefficient of the nozzle. It can now be written as

${e}_{kin}$ | ${\alpha}_{1}$ | ${\alpha}_{2}$ | K |
---|---|---|---|

(24) | 1.088 | 1.069 | −0.125 |

(24) and $k=0$ | 1.077 | 1.060 | 0.018 |

$={u}_{m}^{2}/2$ | 1.0 | 1.0 | 0.897 |

## 5. Nozzle Optimization

#### 5.1. Numerical Details

`OpenFOAM`version 1.6, see [32]. The structured grid exploits axial symmetry, which is applied using

`OpenFOAM`’

`s`

`wedge`boundary conditions at the opposing wedge faces. The boundary condition implies a transformation leading to the appropriate gradients in Cartesian coordinates. There is an angle of five degree with only one cell layer in circumferential direction which reduces the grid size by a factor of 1/72 compared to a corresponding full grid.

`OpenFOAM`’

`s`

`blockMesh`utility with a constant radius in streamwise direction to be rescaled radially by a Matlab script during the optimization. This can be achieved easily, since all point coordinates are stored as a compressed ASCII file. A typical grid for a diffuser is shown in Figure 7.

`OpenFOAM`solver application

`simpleFoam`, which uses the SIMPLE algorithm and computes, after an in-house adaption, the integral of (12) in every outer iteration of the coupled solution procedure. This value is needed to indicate convergence. Convergence is met when the relative difference of $\dot{S}$ between two consecutive iterations is smaller than 1×10

^{-7}and the relative difference of the averaged inlet pressure is smaller than 1×10

^{-5}at the same time. Using residuals as a convergence criterion instead is no option since the correlation between the residuals and changes in $\dot{S}$ can hardly be predicted.

`OpenFOAM`schemes

`Gauss linear`for the convective terms,

`Gauss linear corrected`for the diffusive terms and

`Gauss linear`for the gradients.

#### 5.2. Nozzle Optimization for the Polynomial Wall Shape, $\mathbf{K}=\mathbf{K}({\widehat{X}}_{1},{\widehat{X}}_{2})$

**Figure 8.**Geometry of the polynomial shaped nozzle, here shown for $L/{D}_{1}=1$, i.e., ${\widehat{X}}_{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.75/3.75=0.2$.

**Figure 9.**K-values for $\mathbf{Re}=20000$ in the nozzle design space. (a) Surface plot $\mathbf{K}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{K}(L/{D}_{1},{R}_{m}/{D}_{1})$; (b) $\mathbf{K}=\mathbf{K}({R}_{m}/{D}_{1})$ for fixed values of $L/{D}_{1}$.

**Figure 10.**Cross sectional entropy generation rates for the nozzles; $\mathbf{Re}=20000$; (a) $L/{D}_{1}=0.75$; (b) $L/{D}_{1}=4$; see Figure 8 for symbols.

**Figure 14.**Comparing straight and polynomial nozzles’ cross sectional entropy generation rates; $L/{D}_{1}=1.8$; $\mathbf{Re}=20000$.

## 6. Diffuser Optimization

#### 6.1. Diffuser Optimization for the Polynomial Wall Shape; $\mathbf{K}({\widehat{X}}_{1},{\widehat{X}}_{2})$

**Figure 16.**K-values for $\mathbf{Re}=40000$ in the diffuser design space. (a) Surface plot $\mathbf{K}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{K}(L/{D}_{1},{R}_{m}/{D}_{1})$; (b) $\mathbf{K}=\mathbf{K}({R}_{m}/{D}_{1})$ for fixed values of $L/{D}_{1}$.

**Figure 17.**Cross sectional entropy generation rates for the diffusers; $\mathbf{Re}=40000$; (a) $L/{D}_{1}=4$; (b) $L/{D}_{1}=11.6$; see Figure 8 for symbols.

#### 6.2. Diffuser Optimization for a Non-Straight Wall Shape; $\mathbf{K}({\widehat{X}}_{1},\dots ,{\widehat{X}}_{6})$

`Matlab`, see [33]. This is convenient since

^{Ⓡ}`Matlab`is used to generate the radially transformed structured grid for the repetitive computations which are performed using

^{Ⓡ}`OpenFOAM`, see Section 5.1.

`Matlab Global Optimization Toolbox`is used, which offers a large variety of user settings. The population size is set to be 12 individuals, i.e., twice the number of the individual geometry parameters, which represent the “genome” of each individual. At every optimization step the population consists of one “elite child” from a last iteration, six “crossover children” and five “mutant children”. The crossover children are based on a linear extrapolation of two parents selected from the preceding generation by a roulette wheel selection. Here, the child’s genome lies on a straight line, which crosses the genomes of both parents, at a prescribed distance of 20% away from the better parent in outward direction. This choice seems reasonable, when a comparably small population is used, since a single point crossover might lead to many ill-designed diffusers with corrugated walls. The mutant children are gained from one parent each by replacing each of its six parameters with a fixed probability of 5% by a random value which is bounded by the upper and lower limit $0\le {\widehat{X}}_{i}\le 1$ for this parameter. This method is chosen to introduce new design candidates at every step of the optimization to prevent premature convergence due to the smooth crossover designs.

**Figure 20.**Optimum diffuser design with a PCHIP interpolation of the wall shape with six radii provided at equidistant positions on the centerline for $\mathbf{Re}=100000$.

**Figure 21.**Distribution of the cross sectional entropy generation rates for the optimal and the straight diffuser; $\mathbf{Re}=100000$.

**Figure 22.**Visualization of the investigated design space in parallel coordinates (only elite individuals from every generation are shown); $\mathbf{Re}=100000$.

**Figure 23.**Fluctuating part ${\left({\dot{S}}^{\prime \prime \prime}\right)}^{\prime}$ of the entropy generation rates ${\dot{S}}^{\prime \prime \prime}$; light: high values, nonlinear scale.

## 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

**AMA Style**

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 Style**

Schmandt, 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