March 31, 2022

Approximation models for hydraulic components

Industry: Industrial Equipment | Product: pSeven | Company: Institute of Nuclear Power Engineering and Technical Physics

Problem Statement

Ejectors have many applications across industries. Let us consider a jet pump consisting of a motive fluid inlet, a throat with three suction pipes, and an outlet (Fig. 1). The problem under consideration can be reduced to finding the relation between the suction pipes flow rates and hydraulic resistances, the throat hydraulic resistance, and the motive fluid flow rate.

Fig. 1. Problem statement

Challenges

CFD simulation is conventionally used to solve such type of problems. However, the conventional approach is not suitable for an iterative simulation of an entire hydraulic system containing the jet pump. The reason is that ~1,000 iterations with CFD simulation at each iteration would take too much time. So, an approximation model of the device should be used instead. For this, the platform needs to have interfaces to connect to the CAE software (including in-house solutions), embedded multidimensional approximation tools, and convenient tools for simulation model quality assessment.

Solution

To build the approximation model, first, we generate a K-dimensional (K=12 in this case, including extra variables) training dataset with the DSE block (Fig. 2). The DSE block generates values for the pressure P, motive fluid flow rate Q temperature T and the suction pipe pressures P_1-3. The ConstraintP block monitors that the condition P₁>P₂>P₃ is met. This condition represents the requirement of the upper-level system where the jet pump operates. The Properties block estimates the thermodynamic properties of water from the input variables. The variables produced by the DSE block and the fluid properties from the Properties block are input data for the CFD block which simulates the jet pump hydraulic performance in the STAR-CCM+ simulation software. The ConstraintDP block validates the analysis by checking the estimated hydraulic resistance.

Fig. 2. Training dataset generation

For convenience, the dataset can be represented as a scatter plot (Fig. 3).

Fig. 3. Training dataset. Scatter and correlations plot

The analysis of the cross-correlation factors used for the approximation of the ejector identifies the following statistical dependencies:

(1)

Where u₁ and u₃ are the flow velocities in suction pipes 1 and 3, respectively. These variables were introduced to compensate for the poor correlation between ξ_1,ξ₃ and the input data, m/s. 

The hydraulic resistance factor ξ of the jet pump can be calculated with analytical expression:

 

(2)

Where

• ΔP is the pressure difference between the inlet and outlet of the nozzle, [Pа];
• p is the fluid density, [kg/m³];

• d is the hydraulic diameter to which the ξ, values are normalized, [m].

  To train the approximation model (1) we used several methods as follows. For ΔP and ξ₂ response surface model (RSM);  u₁  - Gaussian process, (GP);  ξ₁, u₃, ξ₃: High-Dimensional Approximation Gaussian Process, (HDAGP). Fig. 4 shows the workflow for ejector performance prediction with approximation models in pSeven.

Fig. 4 Ejector surrogate model inference workflow

Results

Fig. 5 shows the hydraulic resistance factors evaluated with equations (1)…(2) vs. the training dataset values obtained by the CFD simulation.

Fig. 5. Total hydraulic resistance factor

Figs. 6-9 show the values estimated with the (1) approximation models in pSeven vs. the CFD simulation results (training dataset). To validate the training of the approximation models, we generated an additional input dataset {Q,T,P,P₁,P₂,P₃} (validation dataset) not identical to the training dataset. Figs. 6…9 also show the values provided by the approximation model (1) for the validation dataset.

Fig. 6. Total nozzle pressure drop

Fig. 7. Suction pipe 1 hydraulic resistance factor

Fig. 8. Suction pipe 2 hydraulic resistance factor

Fig. 9. Suction pipe 3 hydraulic resistance factor

As we can see, the hydraulic resistance factor trends for the validation dataset are similar to the training dataset trends. It proves that the proposed approximation models are valid.

Table 1. Models accuracy metrics