A hybrid artificial neural network—mechanistic model for centrifugal compressor

Abstract

A mathematical model is an important tool for design and optimization of centrifugal compressor. However, owing to the varying compressor speeds and the complexity of the flow dynamics inside the impeller and diffuser, the currently available mechanistic models may yield inaccurate results. The purpose of this paper is to present a hybrid modeling approach for developing a quantitatively accurate model for centrifugal compressor. Two novel hybrid models, that is, additive and multiplicative hybrid models each of which consists of a three-layer back-propagation artificial neural network (ANN) component and a mechanistic component suitably modified to describe the performances of multistage centrifugal compressor, were constructed and compared with the well-developed ANN model. The results from the hybrid models showed better performance compared to the ANN model. Besides, the hybrid models demonstrated much better performance than the pure mechanistic model, and the multiplicative hybrid model, in general, showed better accuracy than that of the additive hybrid model in our case.

Appendices

Appendix 1: Single-stage centrifugal compressor model for system simulation

The model used for the mechanistic component of the hybrid model is an analytical multistage centrifugal compressor mechanistic model based on the model developed by Jiang [1]. This original model was originally developed for surge control design [5] and its response was compared to the measured response from the laboratory in three different classes of experiments [1, 5]. Further, Jiang et al. [1] developed a modified version which is capable of system simulation in the virtual test bed (VTB) computational environment. For easy implementation in the VTB platform, the nonlinear governing equations are discretized in resistive companion form. The main compressor characteristics equation of the model can be described as follows:

$$ \dot{\omega } = \frac{1}{J} \cdot (\tau_{t} - \tau_{c} ) $$

(9)

$$ \omega \cdot \tau_{c} = W_{c} = m \cdot \Updelta h_{\text{ideal}} $$

(10)

$$ \varepsilon = \frac{{p_{02} }}{{p_{01} }} = (1 + \frac{{\eta_{i} (m,U) \cdot \Updelta h_{\text{ideal}} }}{{T_{01} \cdot c_{p} }})^{{{\raise0.7ex\hbox{$\gamma $} \!\mathord{\left/ {\vphantom {\gamma {\gamma - 1}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\gamma - 1}$}}}} $$

(11)

$$ \eta_{i} (m,U) = \frac{{\Updelta h_{\text{ideal}} }}{{\Updelta h_{\text{ideal}} + \Updelta h_{\text{loss}} }} - \Updelta \eta $$

(12)

$$ \Updelta h_{\text{ideal}} = \frac{{W_{c} }}{m} = \sigma \cdot U_{2}^{2} $$

(13)

where ω is rotational speed (rpm), τ _t is compressor torque (Nm), τ _t is drive torque (Nm), J is spool moment of the inertia (kgm²), m is mass flow rate (kgs⁻¹), W _c is power delivered to the fluid (J), Δh _ideal is ideal specific enthalpy delivered to the fluid (Jkg⁻¹), T ₀₁ is inlet temperature (K), c _p is gas velocity enter the impeller (ms⁻¹), Δh _loss is sum of the incidence loss and friction loss in impeller and diffuser, Δη represents others small losses, such as back flow loss and clearance loss.

Ideally, we would have the same energy transfer Δh _ideal for all flow rates. However, due to various losses, the energy transfer is not constant and we now include this in the analysis. The two major losses, expressed as specific enthalpies, are incidence loss in impeller and diffuser, Δh _ii and Δh _id; friction loss in the impeller and diffuser, Δh _fi and Δh _fd.

$$ \Updelta h_{\text{ii}} = \frac{1}{2}\zeta (U_{1} - C_{\theta 1} - \cot \beta_{1b} C_{a1} )^{2} = \frac{1}{2}\zeta \left(U_{1} - \frac{{\cot \beta_{1b} m}}{{\rho_{01A1} }}\right) $$

(14)

$$ \Updelta h_{\text{id}} = \frac{1}{2}\zeta \left(\frac{{\sigma \cdot D_{2} \cdot U_{1} }}{{D_{1} }} - \frac{{\cot \alpha_{2b} \cdot m}}{{\rho_{01} \cdot A_{1} }}\right) $$

(15)

$$ \Updelta h_{\text{fi}} = \frac{{2{\text{fl}}_{i} }}{{D_{i} \rho_{01}^{2} A_{1}^{2} \sin^{2} \beta_{1b} }}m^{2} $$

(16)

$$ \Updelta h_{\text{fd}} = \frac{{2{\text{fl}}_{d} }}{{D_{d} \rho_{01}^{2} A_{1}^{2} \sin^{2} \alpha_{2b} }}m^{2} $$

(17)

where ζ is the shock loss coefficient, U ₁ is the inlet tangential velocity of the impeller (ms⁻¹), C _θ1 is inlet tangential velocity of gas (ms⁻¹), β _1b is blade inlet angle (rad), C _a1 inlet radial velocity of gas (ms⁻¹), ρ ₀₁ is constant stagnation inlet density (kgm⁻³), A ₁ is reference area (m²), σ is slip factor, D ₁ is inlet reference diameter (m), D ₂ is diameter at the impeller tip (m), α _2b is diffuser inlet angle (rad), f is friction loss, l _i is mean channel length of impeller (m), l _d is mean channel length of diffuser (m), D _i is mean hydraulic channel diameter at impeller (m), D _d is mean hydraulic channel diameter at diffuser (m), and l _d is mean channel length of diffuser (m).

Other losses, such as back flow loss Δη _bf, clearance loss Δη _c  = 0.3(l _cl /b), volute loss 0.02 ≤ Δη _v  ≤ 0.05, and losses 0.02 ≤ Δη _d  ≤ 0.07 due to inadequate diffusion, will be taken into account when computing the efficiency of the compressor.

$$ \eta_{i} (m,U_{1} ) = \frac{{\Updelta h_{02} }}{{\Updelta h_{02} + \Updelta h_{\text{loss}} }} - \Updelta \eta_{c} - \Updelta \eta_{\text{bf}} - \Updelta \eta_{v} - \Updelta \eta_{d} $$

(18)

where Δh _loss = Δh _ii + Δh _id + Δh _fi + Δh _fd. More details of the compressor model can be found in the literature [1, 5, 6].

Appendix 2. Type A and type B mechanistic model

In order to validate the rationale of the hybrid model that ANN can be used to improve the accuracy of compressor performance modeling by estimating the unmodeled part in the mechanistic model, some difference is introduced to the mechanistic model when it is used for the mechanistic component of the hybrid model and producing data sets for training ANN, respectively. Then, two different type mechanistic models, type A and type B, are obtained as described earlier in this paper. Mechanistic models may yield inaccuracy results owing to two major reasons: one is that some complexity physical phenomena in the compressor, such as small losses, cannot be reliably described by the mechanistic models; the other one is that some important aeration kinetics parameters, for example, shock loss coefficient ζ, are hard to be accurately determined by empirical method. Even these parameters can be estimated from experimental or historical data, they are likely to have substantial errors. For simulating the two major reasons above, two kinds of difference were introduced to the mechanistic model; first, when the mechanistic model (type A mechanistic model) is used for the mechanistic component of the hybrid model, the small losses: clearance loss, volute loss, and back flow loss are ignored; when the mechanistic model (type B mechanistic model) is used for producing data, all the losses listed in Eqs. (14) to (18) are considered. Second, the values of the important parameters, that is, shock loss coefficient ζ and the reference area A ₁ are different between the type A and type B mechanistic models as shown in Tables 4 and 5.

Table 4 Parameters for the centrifugal compressor mechanistic model type A

Table 5 Parameters for the centrifugal compressor mechanistic model type B