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Research Papers

# Influence of Thermodynamic Models in Two-Dimensional Flow Simulations of Turboexpanders

[+] Author and Article Information
John Harinck

Energy Technology Section, Process and Energy Department, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlandsj.harinck@tudelft.nl

P. Colonna

Energy Technology Section, Process and Energy Department, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlandsp.colonna@tudelft.nl

A. Guardone

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20159 Milano, Italyguardone@aero.polimi.it

S. Rebay

Dipartimento di Ingegneria Meccanica, Università di Brescia, Via Branze 38, 25123 Brescia, Italyrebay@ing.unibs.it

J. Turbomach 132(1), 011001 (Sep 11, 2009) (17 pages) doi:10.1115/1.3192146 History: Received January 14, 2008; Revised April 20, 2009; Published September 11, 2009

## Abstract

This paper presents a quantitative comparison of the effect of using thermodynamic models of various degrees of complexity if applied to fluid-dynamic simulations of turboexpanders operated at conditions affected by strong real-gas effects. The 2D flow field of a standard transonic turbine stator is simulated using the state-of-the-art inviscid ZFLOW computational fluid-dynamic solver coupled with a fluid property library containing the thermodynamic models. The considered thermodynamic models are, in order of increasing complexity, the polytropic ideal-gas (PIG) law, the Peng–Robinson–Stryjek–Vera (PRSV) cubic equation of state, and the highly accurate multiparameter equations of state (MPEoSs), which are adopted as benchmark reference. The fluids are steam, toluene, and R245fa. The two processes under scrutiny are a moderately nonideal subcritical expansion and a highly nonideal supercritical expansion characterized by the same pressure ratio. Using the PIG model for moderately nonideal subcritical expansions leads to large deviations with magnitudes of up to 18–25% in density, sound speed, velocity, and total pressure loss, and up to 4–10% in Mach number, pressure, temperature, and mass flow rate. The PIG model applied to highly nonideal supercritical expansions leads to a doubling of the deviations’ magnitudes. The advantage of the PIG model is that its computational cost is roughly 1/11 (or 1/3 if saturation-checks in the MPEoS are omitted) of the cost of the MPEoSs. For the subcritical expansion, adopting the physically more correct cubic PRSV model leads to comparatively smaller deviations, namely, $<2%$ (toluene and R245fa) and $<4%$ (steam) in all flow parameters, except for the total pressure loss error, which is comparable to that of the PIG model. The PRSV model is reasonably accurate even for the highly nonideal supercritical expansion, for which the errors are at most 4%. The computational cost of the PRSV model is roughly nine times higher than the cost of the PIG model (or twice as high if saturation-checks in the PRSV are omitted). Contrary to low-complexity fluids like water, for complex fluids like toluene and R245fa the deviations in density, speed of sound, and velocity ensuing from the use of the PIG model vary strongly along the isentropic expansions. This invalidates the approach commonly used in practice of correcting the PIG model with a properly chosen constant compressibility factor.

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## Figures

Figure 1

Coarse grid (1302 nodes) of the VKI LS-89 blade. Periodicity allows the computations to be performed using only the top half of the grid shown here.

Figure 2

Mach number distribution for the MUR43 test case (a) and the MUR49 test case (b). Shown are the experimental measurements (◼) from Ref. 50 and the numerical solution on the coarse grid (-⋅-⋅-), the medium grid (— — —), and the fine grid (—).

Figure 3

Representation of the six expansion processes considered in this work in the temperature-specific entropy diagrams of (a) steam , (b) toluene, and (c) R245fa. The inlet conditions are the same in terms of reduced temperature and pressure. For each fluid, one expansion starts from a supercritical state (denoted by 1SC on the diagram) and the other from a subcritical state 1.

Figure 4

Percentage difference of thermodynamic properties (a) sound speed and (b) density along the subcritical expansion isentrope computed based on the PRSV model (◼) and the PIG model (no symbol) with respect to the values computed based on the MPEoS. Results are given for steam (blue line), toluene (red line), and R245fa (green line). (Color representation of this figure is available at ASME.org)

Figure 5

Flow field distributions of steam: (a) Mach number and (b) pressure coefficient distribution and streamlines based on the MPEoS; (c) Mach number distribution and streamlines based on the PIG model

Figure 6

Blade surface distributions for steam of (a) the isentropic Mach number, (b) pressure coefficient, (c) sound speed, (d) velocity, (e) density, (f) compressibility factor, (g) fundamental gasdynamic derivative, and (h) temperature, computed based on the MPEoS (—), PRSV (- - -), and PIG (-⋅-⋅-) model

Figure 7

Percentage difference of (a) the isentropic Mach number, (b) pressure, (c) sound speed, (d) velocity, (e) density, and (f) temperature, along the blade surface, computed based on the PRSV model (◼) and the PIG model (no symbol) with respect to the values computed based on the MPEoS. Results are given for steam (blue line), toluene (red line), and R245fa (green line). The blade suction and pressure sides are distinguished by the continuous and dashed lines, respectively. (Color representation of this figure is available at ASME.org)

Figure 8

Expansion isentropes as obtained using the MPEoS (—), PIG (- - -), and ZPIG (-⋅-⋅-) models in the pressure-specific volume diagrams (showing the saturation line) of steam (a), toluene (b), and R245a (c). In the ZPIG model, the compressibility factor Z is set to the value given in Table 4. The dotted intersection of MPEoS and ZPIG indicates the subcritical inlet state.

Figure 9

Flow field distributions of toluene: (a) Mach number and (b) pressure coefficient distribution, and streamlines based on the MPEoS; (c) Mach number distribution and streamlines based on the PIG model

Figure 10

Blade surface distributions for toluene of (a) the isentropic Mach number, (b) pressure coefficient, (c) sound speed, (d) velocity, (e) density, (f) compressibility factor, (g) fundamental gasdynamic derivative, and (h) temperature, computed based on the MPEoS (—), PRSV (- - -), and PIG (-⋅-⋅-) model

Figure 11

Flow field distributions of R245fa: (a) Mach number and (b) pressure coefficient distribution and streamlines based on the MPEoS; (c) Mach number distribution and streamlines based on the PIG model

Figure 12

Blade surface distributions for R245fa of (a) the isentropic Mach number, (b) pressure coefficient, (c) sound speed, (d) velocity, (e) density, (f) compressibility factor, (g) fundamental gasdynamic derivative, and (h) temperature, computed based on the MPEoS (—), PRSV (- - -), and PIG (-⋅-⋅-) model

Figure 13

Supercritical expansion: percentage difference of (a) the isentropic Mach number, (b) pressure, (c) sound speed, (d) velocity, (e) density, and (f) temperature, along the blade surface, computed based on the PRSV model (◼) and the PIG model (no symbol) with respect to the values computed based on the MPEoS. Results are given for toluene (red line) and R245fa (green line). The blade suction and pressure sides are distinguished by the continuous and dashed lines, respectively. (Color representation of this figure is available at ASME.org)

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