An Analytical Technique of Transient Phase-Change Material Melting Calculation for Cylindrical and Tubular Containers

ABSTRACT

In this study, an analytical model for a class of heat storage that utilizes latent heat of a phase-change material (PCM) is developed. Two basic shell-and-tube configurations are considered, one in which the PCM melts inside the tubes while the heat transfer fluid (HTF) flows in the shell along it, and the other in which HTF flows inside the tubes while PCM melts outside. A system of partial differential equations, which describes heat transfer and melting of the PCM and heat transfer in the HTF, is derived with some simplifying assumptions, while still capturing and preserving the essential features of the processes involved. These equations are solved analytically, yielding the overall heat exchange parameters, like instantaneous heat transfer rate, stored energy, and overall operation time of the system. The present work shows that the use of the proposed analytical technique and its modifications for the practical PCM arrangements is beneficial. Proper application of the model makes it possible to obtain the parameters of a real PCM melting process in the form of algebraic formulas, both for the transient values of variables over time, and for the overall process characteristics. A comparison with the results of numerical calculations of transient melting, made using computational fluid dynamics, confirms the validity of analytical findings and allows to assess the degree of accuracy of the results of our analytical method in various practical cases.

Nomenclature

A	=	heat transfer area, related to diameter D, m2
b	=	constant value, dimensionless, (Eq. 14(14) $$\begin{array}{ccc}\hfill b& =& log\left[1+h\frac{D}{4k}log\left(1+\mathit{w}\right)\right]\left(\mathrm{pipe}\right)\hfill \\ \hfill b& =& log\left[1+h\frac{D}{k}\right]\left(\mathrm{cylinder}\right)\hfill \end{array}$$(14) )
b1	=	constant value, dimensionless, (Eq. 27(27) $$\begin{array}{c}\hfill \frac{d{f}_{m0}}{f_{m0}\left\{1-f_{m0}\left[1-exp\left(-b\right)\right]\right\}}=\frac{b_1{h}_0}{h_f}dy\end{array}$$(27) )
b2	=	constant value, dimensionless, (Eq. 36(36) $$\begin{array}{c}\hfill b_2=exp\left(b_1\frac{h_0}{h_f}\right)-1\end{array}$$(36) )
cp	=	HTF specific heat capacity, J/kg/K
cpp	=	PCM specific heat capacity, J/kg/K
cpt	=	tube material specific heat capacity, J/kg/K
CFD	=	computational fluid dynamics
D	=	tube diameter related to PCM, m
Ds	=	inner diameter of outer shell-tube, m
dm	=	effective diameter of a PCM solid–liquid border, m
fm	=	PCM melt fraction, dimensionless
fm0	=	PCM melt fraction in inlet cross-section, dimensionless
h	=	heat transfer coefficient from HTF to PCM, W/m2/K
h0	=	constant value of heat transfer coefficient dimension, W/m2/K
hf	=	constant value of heat transfer coefficient dimension, W/m2/K
HTF	=	heat transfer fluid
k	=	PCM thermal conductivity, W/m/K
L	=	PCM latent heat, J/kg
Leff	=	effective PCM latent heat, J/kg
M	=	PCM mass, kg
Mt	=	mass of tube material, kg
$\dot{m}$	=	HTF mass flow rate, kg/s
p	=	constant value, dimensionless, (Eq. 11(11) $$\begin{array}{c}\hfill p=h_0\frac{D}{4k}\end{array}$$(11) )
PCM	=	phase-change material
Q	=	absorbed heat, J
Q0	=	overall absorbed heat, J
q0	=	overall heat transfer rate, W
qmax	=	maximum overall heat transfer rate, W
q'	=	heat transfer rate per unit length, W/m
q”	=	heat transfer rate per unit area, W/m2
r	=	thermal resistance of the annular PCM liquid layer, per W m2 K
T	=	HTF temperature, °C
Tin	=	inlet HTF temperature, °C
Tm	=	PCM melting temperature, °C
t	=	time, s
ti	=	time of complete melting of inlet cross-section, s
v	=	HTF velocity, m/s
w	=	constant value, dimensionless, (Eq. 3(3) $$\begin{array}{ccc}\hfill r& =& \frac{D}{4k}log\left(1+\mathit{w}{f}_m\right)\left(\mathrm{pipe}\right)\hfill \\ \hfill r& =& -\frac{D}{4k}log\left(1-f_m\right)\left(\mathrm{cylinder}\right)\hfill \end{array}$$(3) )
X	=	length of PCM system, m
x	=	current longitudinal coordinate, m
y	=	auxiliary function, dimensionless, (Eq. 25(25) $$\begin{array}{c}\hfill y\left(x,\tau \right)=\left\{\begin{array}{c}\frac{h_f\left[1-exp\left(-b\right)\right]}{h_0}\left\{\tau +\frac{1}{b}log\left[1-exp\left(-b\tau \right)\right]\right\}\hfill \\ -x/X,\tau \le 1\hfill \\ \frac{h_f\left[1-exp\left(-b\right)\right]}{h_0}\left\{1+\frac{1}{b}log\left[1-exp\left(-b\right)\right]\right\}\hfill \\ +\frac{h_f}{h_0}\left(\tau -1\right)-x/X,\tau \ge 1\hfill \end{array}\right.\end{array}$$(25) )

Greek Symbols
θ	=	auxiliary function of time, dimensionless, (Eq. 35(35) $$\begin{array}{c}\hfill \theta \left(\tau \right)=1+exp(-b)\left\{exp\left(b_1\left[\frac{h_0}{h_f}-\left(\tau -1\right)\right]\right)-1\right\}\\ \hfill \varphi \left(\tau \right)=1-exp\left(-b\right)\frac{h_f}{h_0}\left(\tau -1\right)\end{array}$$(35) )
τ	=	dimensionless time
τpc	=	time of complete melting of particular cross-section, dimensionless
τ0	=	overall time of complete melting, dimensionless
τ0min	=	overall time of complete melting, minimal value, dimensionless
φ	=	auxiliary function of time, dimensionless, (Eq. 35(35) $$\begin{array}{c}\hfill \theta \left(\tau \right)=1+exp(-b)\left\{exp\left(b_1\left[\frac{h_0}{h_f}-\left(\tau -1\right)\right]\right)-1\right\}\\ \hfill \varphi \left(\tau \right)=1-exp\left(-b\right)\frac{h_f}{h_0}\left(\tau -1\right)\end{array}$$(35) )
ϕ	=	auxiliary function of time, dimensionless, (Eq. 23(23) $$\begin{array}{c}\hfill \phi \left(\tau \right)=\left\{\begin{array}{c}\frac{h_f\left[1-exp\left(-b\right)\right]}{h_0}\left\{\tau +\frac{1}{b}log\left[1-exp\left(-b\tau \right)\right]\right\}\hfill \\ +C,\tau \le 1\hfill \\ \frac{h_f}{h_0}\tau +C_1,\tau \ge 1\hfill \end{array}\right.\end{array}$$(23) )

Additional information

Notes on contributors

Vadim Dubovsky

Vadim Dubovsky, Ph.D., is a Research Engineer in the Heat Transfer Laboratory, Mechanical Engineering Department, Ben-Gurion University of the Negev, Israel. He has been extensively involved in various research activities in the fields of passive ventilation, thermal storage, and applications of phase-change materials.

Gennady Ziskind

Gennady Ziskind is Professor and Head of the Heat Transfer Laboratory, Mechanical Engineering Department, Ben-Gurion University of the Negev, Israel. His main fields of interest are heat storage, thermal management, and multiphase systems. He was the Conference co-Chair (with Prof. Luisa F. Cabeza) of Eurotherm Seminar 99: Advances in Thermal Energy Storage (Lleida, Spain) in May 2014. He serves as an Associate Editor of ASME Journal of Heat Transfer.

Ruth Letan

Ruth Letan is Professor Emeritus in the Mechanical Engineering Department, Ben-Gurion University of the Negev, Israel. Present research: PCM in thermal systems, wavy fins, multi-layer insulation, and stratification in enclosures.