## A Unified Theory Characterising the Spontaneous Condensation Tendency of Working Fluids in Organic Rankine Cycle Expansion: Key Difference from Steam and its Implications

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## Abstract

The present work formulates a unified theory characterising the spontaneous condensation tendency of various working fluids during a typical Organic Rankine Cycle (ORC) expansion from slightly superheated condition or dry-saturated condition directly into the saturation dome, in the context of an ORC expander. Traditional wisdom on steam Rankine cycles dictates that the expansion process be almost completely outside the saturation dome, owing to wetness loss and blade erosion problems. Current practice of ORC design tends to follow this traditional wisdom. However, the present work finds that organic working fluids, characterised by a molecular size much larger than that of water, are much less likely to condense in a typical ORC expansion into the saturation dome, thereby allowing the expansion process to be situated entirely within the saturation dome without incurring the abovementioned wetness-related problems. This can lead to significant improvement of overall conversion rate of available heat from a finite heat source to useful work, and cycle’s overall power density (defined in either useful power per unit heat exchanger area or useful power per unit initial cost).

Specifically, in the introductory Chapter 1, the author introduces the idea of temperature matching, which led to the definition of the Thermal Matching Number Th, a new nondimensional number that quantifies the temperature matching characteristics of a cycle heating curve with a given finite heat source, with better temperature matching meaning higher overall conversion rate from available heat to useful work. Then it was shown that operating the expansion process of a power generating cycle from dry-saturated state directly into the saturation dome, provided that negligible condensation penalty is incurred, as opposed to the conventional steam Rankine cycle way of superheating followed by an almost completely vapour-phase expansion, is thermodynamically more favourable because it increases the Thermal Matching Number Th of the cycle heating curve and the cycle power density.

In Chapter 3, the author discusses the Ludwieg tube-based experimental methods for pressure measurements of expansion flow through an actual ORC transonic vane geometry. The experimental results serve the purpose of code validation of the two-phase numerical scheme developed in this work in Chapter 4. The same transonic vane geometry is also used for application of the unified theory (characterising various working fluids’ condensation tendency) in an actual ORC turbomachinery setting in Chapter 5.

In Chapter 4, the author describes a two-dimensional time-marching mixed Eulerian-Lagrangian numerical scheme for solving inviscid two-phase condensing flows of various organic working fluids, following a similar strategy to previous mixed Eulerian-Lagrangian wet steam methods developed in the Cambridge University Engineering Department. The numerical scheme solves the two-phase mixture flow in a Eulerian frame of reference, and integrates the nucleation rate and droplet growth equations in an embedded Lagrangian frame of reference, allowing calculations based on the two frames to be decoupled. Five test cases used for code validation, where numerical results are compared against experimental data, gave good agreement and confirmed the numerical accuracy of the numerical method in capturing essential characteristics of the gasdynamics of spontaneously condensing flows and complex transonic vane flows.

In Chapter 5, the author first investigates the reason why the associated nucleation rate of different fluids in typical ORC expansions into the saturation dome are vastly different, while demonstrating that the associated nucleation rate is a fluid specific property that is negatively correlated with the molecular size of the fluid, and emphasising that the nucleation rate forms the root determinant of the difference in condensation tendency of various fluids. Then a unified theory characterising the spontaneous condensation tendency of various working fluids during a typical ORC expansion into the saturation dome is proposed, which includes formulation of two new nondimensional numbers, namely the Condensation Number C and the pressure-based Relative Enthalpy Number Ep. The Condensation Number C directly computes the wetness fraction accumulation generated from nucleation alone, whereas the pressure-based Relative Enthalpy Number Ep calculates the extent of perturbation to isentropic gasdynamics due to heat released by nucleation alone. Derived from completely different fundamental physical considerations, both nondimensional numbers enable the condensation tendency among different fluids to be compared on the same scale, and both from a physical standpoint highlight the importance of nucleation rate in determining the fluid’s condensation tendency. Henceforth, a robust logical connection has been established, from a fluid’s molecular volume, to its associated nucleation rate, and finally to its condensation tendency. An important conclusion then follows: for organic working fluids with relatively large molecular size (compared to water), their associated nucleation rate and therefore condensation tendency in typical ORC expansions is so low that an expansion into the saturation dome from dry-saturated or slightly superheated condition is virtually condensation-free. Without condensation and associated problems, it is therefore viable to situate the ORC expansion process completely inside the saturation dome, thereby taking full advantage of the enhanced Thermal Matching Number Th of the cycle heating curve and enhanced cycle power density, as discussed in Chapter 1.

As R134a and water were found to be excellent examples of fluids with vastly contrasting condensation tendency behaviours in typical ORC expansions, they are further tested in the transonic vane cascade geometry, typical of actual Organic Rankine Cycle applications. C and Ep analyses for a mid-pitch streamline for each test case were performed, with the results confirming the vast difference in condensation tendency between R134a and water in typical ORC expansions in a turbomachinery setting. The results again confirmed that large-molecule low-condensation-tendency fluids remain virtually condensation-free in typical ORC expansions into the saturation dome, suggesting that the thermodynamic benefits offered by a shift from the conventional outside-dome expansion to within-dome expansion in actual ORC applications can be fully taken advantage of.

Finally, Chapter 5 sums it all up by identifying a range of ORC working fluids, capable of within-dome expansion with negligible condensation penalty, while confirming the general trend of negative correlation between a fluid’s condensation tendency in a typical ORC expansion and its molecular volume.

It is acknowledged that the formulation of condensation tendency theory in this work has only been supported through numerical means; though the order-of-magnitude division among the fluids tested in typical ORC expansions is unequivocal, rendering the fine tuning of parameters in the Classical Nucleation Theory (CNT) unnecessary, it has to be admitted that the numerical treatment is not free of uncertainties, for example with metastable-state modelling, or even with CNT itself. This necessitates further experimental investigation, such as that with high-resolution pressure measurements and/or laser-based droplet size measurements, to fully validate the findings of the thesis, for a range of ORC fluids.