Viscosity

Viscosity is a property that is due to the internal friction of the liquid. The unit of viscosity is [kg/m s)] or Pascal-second [Pa s].

The dynamic fluid viscosity η [kg/(m s)] and, in the case of large temperature gradients, the density ρ [kg/m³] are temperature-dependent variables whose change with temperature means that the differential equations (the transient flow and energy transport equations) represent a coupled system of differential equations. An experimental formula to describe the relationship between dynamic fluid viscosity and temperature is used in the calculation:

Dynamic viscosity η of water as a function of temperature at a density of ρ= 1000.0 [kg/m³]

The unit millipascal second [mPa s] corresponds to the unit 1000*[kg/(m s)]. The following relationship applies between dynamic and kinematic viscosity:

dynamic viscosity: η = μ * ρ [kg/(m s)]

kinematic viscosity: μ = η / ρ [m²/s]

With:

η = dynamic viscosity [kg/(m s)]

μ = kinematic viscosity [m²/s]

ρ = density [kg/m³]

Temperature-dependent density

In SPRING, a linear dependence of the density [kg/m³] on the temperature T [°C] is considered:

ρ(T) = ρ₀ + α(T - T₀)

With:

ρ(T) = temperature dependent density [kg/m³]

ρ₀ = density of the reference temperature T₀ in [kg/m³], default is ρ₀ = 1000.0 [kg/m³]:

ρ₀ = ρ(T = T₀)

T₀ = Reference temperature [°C], the default setting is T₀ = 20°C

α = constant density gradient, the default setting is α = 0.0

The following figure shows the dependence of density on temperature:

Temperature-dependent density function

In this case is:

ρ₀ = ρ(T₀=20 °C) = 1010,5 [kg/m³]

The density function is approximated in the temperature interval under consideration (here: approx. 32 - 75 °C) by a straight line equation (green line), which defines the density gradient α.

In the example, the density gradient α = constant = -0.4756. (This corresponds to the gradient (= 1st derivative) of the straight line:

If the initial or calculated temperatures are in a different temperature range, it may be necessary to use a different temperature range to define another equalisation line (red line).

Realization in SPRING