Supersonic jet pumps are devices that are able to pump a flow without the need for moving parts. The flow pressure variation is achieved mixing the flow (also called *entrained flow*) with a high velocity jet (or *primary flow*) and making them go through a duct of variable cross-sectional area (Eves et al. 2011). As a result of the lack of moving parts, supersonic jet pumps boast a far longer life compared to other pump typologies and for this reason they can be considered environmentally friendly devices. Their application is common in refrigeration to desalination industry (Eves et al. 2011).

Eves et al. 2011 applied a formal optimisation framework to the optimisation of a supersonic jet pump design. The entrained flow rate V_ent was maximised for a broad range of primary flow rates V_pri (from 200 L/min to 1200 L/min). For the application of the formal optimisation framework, the problem was reformulated as follows:

maximise:

V_ent (6.10)

subject to:

V_pri <= c_0 (6.11)

c_i_L <= DV_i <= c_i_U i = 1,2,3 (6.12)

the entrained flow rate V_ent and the primary flow rates V_pri were assumed function of the independent variables X, Y, Z defined as:

X = DV1 = R_N

Y = DV2 = R_DI / R_N

Z = DV3 = R_D0 / R_DI

where R_N is the jet pump nozzle radius, R_DI the diffuser inlet radius and R_D0 the diffuser outlet radius. The input variables Y and Z were defined as ratios of physical jet pump parameters to avoid unfeasible designs.

CFD simulations provided the values of the objective V_ent and the constraint V_pri for each point of a Latin Hypercube DoE made of 100 points. Moving Least Squares Method (MLSM) was used to generate metamodels of the entrained and primary flow rate and GA and SQP algorithms were used to search for the solution of the optimisation problem defined in Eqs. 6.10-6.11-6.12).

**Entrained flow modelling using HyGP**

The idea to use HyGP to model the jet pump entrained flow rate sparked from a practical need: the final user of the jet pump model required a simple tool to evaluate the jet pump performance for a given set of design parameters. Genetic programming models are returned as explicit text expressions that can be evaluated by spreadsheet software, so HyGP was in this case considered the optimal modelling tool. The request of the jet pump user then gave the opportunity to assess HyGP applicability to industrial problems, as well as confirming that accuracy is not the only criterium that guides the selection of a metamodelling technique.

The modelling stage was repeated using HyGP instead of MLSM on the already existing data kindly provided by Dr. Eves. Model generation and validation were performed on the existing building and validation data sets, respectively made of 136 and 57 points sampled in the following region:

1.0 mm <= X <= 3.38mm (6.16)

2.0 <= Y <= 3.5 (6.17)

1.5 <= Z <= 4.5 (6.18)

The minimum distance between DoE points in the building and validation data sets are shown in Fig. 6.13. For each point the corresponding entrained flow rate had been previously computed by CFD simulations.

Fig 6.13 : Minimum distance bewteen DoE points for building and validation data sets

10 HyGP independent runs were performed, setting the population size to 200 individuals and the maximum number of generations to 50. The best model found is reported in Eq. 6.19:

V_ent(X, Y, Z) = 395.18 X – 602.739 Y – 764.272 Z + 969.842 XY + 169.974 YZ – 2892.42 X/Y + 160.407 Y/X – 300.893 (X^2)Y – 9.69909 (X^3)Y – 62.4153 X/Z + 924.589 Z/Y +

639.652 X^2 – 5.50577 (X^2)(Y^2) + 980.552/(XZ) + 439.244 (X^2)/Z + 34.079 (X^3)/Z +

0.528353 X(Y^3)(Z^2) – 737.5 (X^2)/(YZ) – 171.104 (Y^2)/((X^2)Z) + 81.8517 (Y^2)/((X^3)Z) –

304.829 XY/Z – 7.1222 X(Y^3)Z + 36.2463

The RMSE, coefficient of determination R2 and the maximum relative error are provided in Table 6.5. The metamodel features good generalisation properties as the RMSE and R2 on building and validation data sets are comparable.

Table 6.5 : Entrained flow rate metamodels quality on building and validation data sets

Parameter Building Validation

RMSE 10.0131 11.3879

R2 0.998373 0.997664

Max relative error 20.1211 22.4671

In Fig. 6.14 the estimated entrained flow rate is plotted against the actual flow rate computed by CFD for each sample in the building and validation data sets: the points closeness to the line reflects the high accuracy of the metamodel on both data sets.

Fig 6.14 : Model response versus actual response.

Each point represents an entrained flow rate in L/min

The accuracy of the model was considered acceptable for industrial use, and the availability of the explicit expression in Eq. (6.19) made metamodel exploitaiton easer. For example, the plot in Fig, 6.15 showing the dependency of the entrained flow rate on the three input variables was generated using the expression returned by HyGP. In the figure it can be clearly seen that the entrained flow rate maximum lies approximately in the region centred in (X, Y, Z) = (3.3, 2.4, 1.5), fact that is consistent with the results reported by Eves et al. (2011).

Figure 6.15: Entrained flow rate (L/min) as a function of X=DV1, Y=DV2, Z=DV3

**References**

- J. Eves, V. V. Toropov, H. M. Thompson, N. Kapur, J. Fan, D. Copley and A. Mincher,

“Design optimization of supersonic jet pumps using high fidelity flow analysis”,*Structural and Multidisciplinary Optimization,*2011