**Abbreviations**


#### 1. Churchill, S.W.; Chu, H.H.S. Correlating equations for laminar and turbulent free convection from a horizontalcylinder. *Int. J. Heat Mass Transf.* **1975**, *18*, 1049–1053. [CrossRef]


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*Article*

## **Semi-Analytical Model for Heat and Mass Transfer Evaluation of Vapor Bubbling**

#### **Giuseppe Starace, Lorenzo Carrieri and Gianpiero Colangelo \***

Department of Engineering for Innovation, University of Salento, 73100 Lecce, Italy; giuseppe.starace@unisalento.it (G.S.); carrierilorenzo@gmail.com (L.C.)

**\*** Correspondence: gianpiero.colangelo@unisalento.it; Tel.: +39-0832-299440

Received: 30 January 2020; Accepted: 25 February 2020; Published: 2 March 2020

**Abstract:** Multi-stage refrigeration systems cover a wide range of possibilities and are diffusing more and more. The idea that inspired this work derived from the need to have a tool to model the energy behavior of the intercooler inside a multi-stage refrigeration system. In this work, a semi-analytical model of a single bubble, injected into the liquid of an intercooler of a multi-stage system, has been developed. The developed model is a set of equations derived from the Fourier equation for heat conduction in defined conditions and includes the effects of sensible and latent heat. The vapor bubble is supposed to be injected in the saturated liquid contained in a tank at a defined depth, at an intermediate pressure. The model has been implemented in Matlab and the results show the influence of the liquid surface tension, the injection depth and the thermal diffusivity of the vapor. The model developed here is a useful low-cost tool for evaluating heat transfer optimization of a separator/intercooler of a multi-stage refrigeration system.

**Keywords:** Semi-analytical model; vapor; liquid; bubble; two-phase heat transfer
