Transport and exchange equation

Water temperature in the network is a fonction of upstream temperature and thermal exchange with atmosphere and substratum.

The general equation is:

$\frac{\partial ST}{\partial t}+ \frac{\partial QT}{\partial x}= \frac{\partial}{\partial x}\left(DS\frac{\partial T}{\partial x}\right) + SE_T$

where $E_T$ is an exchange rate (°C/s) given by the heat exchange rate $P$ (W/m²). For a water colomn with a width of $\delta x$, energy balance is

$\rho S \delta x \ c_p E_T = P L_m \delta x$

where $\rho$ is the mass density of water, $c_p$ is the heat capacity of water, $L_m$ is the reach’s width, $P$ is the heat power per unit area, yielding

$E_T = \frac{L_m P}{\rho c_p S}$

Numerical values are :

$\rho$ = 1000 kg/m³

$c_p$ = 4186 J/kg/°C

Calculation of the heat power term

The heat power per unit area is a balance of direct incoming solar and atmospheric radiation, minus the water surface radiation, and sensible and latent heat flow. The heat exchange between water and soil is difficult to evaluate since it depends on the soil’s temperature and thermal parameters.

A mask coefficient $C_m$ is defined as a function of abscissa, which takes into account the potential coverage of the canal (passage in galleries in particular). Full coverage will cancel all exchange between water and atmosphere. Neglecting interactions with the substrate amounts to considering the environment as adiabatic or to assume that these exchanges are low with respect to the convection process of the water mass.

The heat power term is thus calculated :

$P = (1 - C_m)((1 - a) R_N + R_a - R_e - H_s - H_e)$

Solar radiation is a measured data that can be estimated using the solar radiation above atmosphere, the atmospheric attenuation and the time and location.Albedo $a$ is generally low for water, around 0.03.

Atmospheric radiation is given by Stefan Boltzmann formula, and takes into account water reverberation, clouds and air relative humidity ($w_a$, between 0 and 1) using Brutsaert formula (1982) [1]. We therefore need to calculate the equilibrium vapor pressure $e_s$ (Pa) and vapor pressure $e_e$ (en Pa) at air temperature $T_a$ (°C) with a humidity $w_a$:

$e_s = 101300 \exp \left (13.7 - \frac{5120}{273.15 + T_a}\right )$

$e_{a} = w_{a} e_{s}$

$c_a = 1,24 (1 - a)\left (\frac{e_a / 100}{273.15 + T_a}\right )^{1 / 7}$

$R_a = c_a \sigma (273.15 + T_a)^4$

where $\sigma = 5,67 10^{-8}$ W m^-2K^-4 is Stefan-Boltzmann constant.

Similarly, water radiation is :

$R_e = \epsilon \sigma (273.15 + T_e)^4$

where $T_e$ is the water temperature (°C) and $\epsilon$ is water emissivity ($\epsilon=0,97$).

Sensible heat flow is linked to thermal gradient between air and water and to the wind speed $U_V$.
We use De Bruin’s formula (1978), in W/m$^2$/Pa, as cited in [2]:

$f_V =0.029+0.021 U_V$

and

$H_s = C_B f_V (T_e - T_a)$

where $C_B = 63$ Pa/° C (Bowen coefficient).

Then, latent heat flow is calculated as follows :

$H_e = f_V (e_s - e_a)$