Mathematical formulation

The fluid flow and heat transfer phenomena involved in processes of water storage tanks are described by the Navier-Stokes and energy equations. Assuming a Newtonian fluid behaviour, with constant physical properties with the exception of density variations which are treated assuming Boussinesq approximation (relevant in buoyancy terms of momentum equations), viscous dissipation and the influence of pressure in temperature negligible and non-participant radiation medium, the governing equations can be written as follows:

V • V = 0

(i)

^ + pv ■ = — Vp + v • T — рЯіЗ (T — To)

(2)

dT -► f к

(3)

PW + Pv. m=v-(-vi)

where і is time; p mass density; v velocity; f stress tensor that is evaluated considering Stokes’ law; p pressure; <f gravity; temperature; T0 reference temperature; ер specific heat at constant pressure; & thermal conductivity; and /3 thermal expansion coefficient. Thermo­physical properties considered in this work are listed in table 1.

Table 1: Thermo-physical properties. Units in SI.

Property

Material

Water

Plexiglass

p

1000

262

Cp

4169

1050

к

0.5552

0.17

P

9.32 10“4

/3

2.76 10-‘1

Reproducing the test sequence proposed above, at the beginning the tank temperature is set at 20oC, which corresponds to preconditioning phase (P1).The inlet mass flow rate has been imposed according to those recommended by the test sequence. Thermal losses of the tank have been modelled considering a heat transfer coefficient of 3 W/m2K at lateral walls, and at the top and bottom of the tank. Ambient temperature has been fixed at 20oC. At the outlet, the injected flow rate has also been imposed, and temperature derivative has been assumed null.