Physical Models

Physical properties and correlations are the most variant and rapidly changing code chunks. This package contains different options for the following physical phenomena (and yielded parameters):

• Hydraulic Friction (friction coefficient, f)

• Single Phase Heat Transfer (heat transfer coefficient, htc)

• Reactor Safety Thermohydraulic Thresholds

• Residual Heat (power contributions, P)

• Heat Transfer Coefficient
• Pressure Drop
• Thresholds
• Decay Heat

Dimensionless Numbers

Several flow properties and equations are given here, including dimensionless parameters and corresponding correlations

Gr(rho, mu, beta, T, Twall, Dh, g=9.80665)

The Grashof number (Gr) is an approximation of the ratio of buoyancy to viscous forces.

For vertical flat plates:

\[\text{Gr} = \rho^2 g\beta (T_\text{wall}-T_\text{bulk})\frac{L_h^3}{\mu^2}\]

Parameters:

• rho (KgPerM3) – fluid density

• mu (PaS) – dynamic viscosity of the fluid

• beta (PerC) – fluid thermal expansion coefficient

• T (Celsius) – fluid bulk temperature

• Twall (Celsius) – wallsurface temperature

• Dh (Meter) – Hydraulic diameter

• g (MPerS2) – Gravitational acceleration constant

Returns:

Gr – The Grashof Number

Return type:

Value

Examples

>>> Gr(rho=1, mu=1, beta=1, T=50, Twall=50, Dh=1)
0.0

Nu(h, L, k)

The Nusselt number (Nu) is defined as the ratio between convective and conductive heat transfer across the boundary. \(\text{Nu} = hL/k\)

Parameters:

• h (WPerM2K) – heat transfer coefficient

• L (Meter) – characteristic or equivalent length

• k (WPerMK) – thermal conductivity of the fluid

Returns:

Nu – The Nusselt number

Return type:

Value

Examples

>>> Nu(h=1., L=1., k=1.)
1.0

Pe(rho, v, L, cp, k)

The Peclet number (Pe) is defined as the ratio between advective and diffusive transport rates. For heat transfer, it is the product of \(\text{Pe} = \text{Re}\text{Pr}\)

Parameters:

• rho (KgPerM3) – fluid density

• v (MPerS) – characteristic or equivalent velocity

• L (Meter) – characteristic or equivalent length

• cp (JPerKgK) – specific heat of the fluid

• k (WPerMK) – thermal conductivity of the fluid

Returns:

Pe – The Peclet Number

Return type:

Value

Examples

>>> Pe(rho=1., v=1., L=1., cp=1., k=1.)
1.0

Pr(cp, mu, k)

The Prandtl number (Pr) is defined as the ratio between viscous diffusivity and thermal diffusivity. \(\text{Pr} = c_p \mu / k\)

Parameters:

• cp (JPerKgK) – specific heat of the fluid

• mu (PaS) – dynamic viscosity of the fluid

• k (WPerMK) – thermal conductivity of the fluid

Returns:

Pr – The Prandtl number

Return type:

Value

Examples

>>> Pr(cp=0.5, mu=0.1, k=50)
0.001

Ra(rho, mu, cp, k, beta, T, Twall, Dh, g=9.80665)

The Rayleigh number (Ra) is associated with heat transfer for natural convection. At low values, heat transfer is primarily conductive, and at high values it is primarily convective. \(\text{Ra} = \text{Gr}\text{Pr}\)

Parameters:

• rho (KgPerM3) – fluid density

• mu (PaS) – dynamic viscosity of the fluid

• cp (JPerKgK) – specific heat of the fluid

• k (WPerMK) – thermal conductivity of the fluid

• beta (PerC) – fluid thermal expansion coefficient

• T (Celsius) – fluid bulk temperature

• Twall (Celsius) – wallsurface temperature

• Dh (Meter) – Hydraulic diameter

• g (MPerS2) – Gravitational acceleration constant

Returns:

Ra – The Rayleigh Number

Return type:

Value

Examples

>>> Ra(rho=1, mu=1, cp=1, k=1, beta=1, T=50, Twall=50, Dh=1)
0.0

Re(rho, u, L, mu)

The Reynolds number (Re) is defined as the ratio between inertial forces and viscous forces. \(\text{Re} = \rho u L / \mu\)

Parameters:

• rho (KgPerM3) – fluid density

• u (MPerS) – characteristic or equivalent velocity

• L (Meter) – characteristic or equivalent length

• mu (PaS) – dynamic viscosity of the fluid

Returns:

Re – The Reynolds number

Return type:

Value

Examples

>>> Re(rho=0., u=1., L=1., mu=1.)
0.0
>>> Re(rho=1., u=np.arange(-2, 3), L=1., mu=1.)
array([2., 1., 0., 1., 2.])
>>> Re(rho=1., u=1., L=1., mu=np.inf)
0.0

Re_mdot(mdot, A, L, mu)

The Reynolds number (Re) is defined as the ratio between inertial forces and viscous forces. If \(\dot{m} = \rho uA\) is known, one may write \(\text{Re} = \dot{m}(L / A) / \mu\)

Parameters:

• mdot (KgPerS) – fluid mass current

• A (Meter2) – flow area

• L (Meter) – characteristic or equivalent length

• mu (PaS) – dynamic viscosity of the fluid

Returns:

Re – The Reynolds number

Return type:

Value

Examples

>>> Re_mdot(mdot=1., A=1., L=1., mu=1.)
1.0
>>> Re_mdot(mdot=-1., A=1., L=1., mu=1.)
1.0
>>> Re_mdot(mdot=0., A=1., L=1., mu=1.)
0.0

flow_regimes(re, bounds)

Turbulent, Laminar and Interim regimes are determined by the Reynolds No.

Parameters:

• re (np.ndarray) – Reynolds numbers vector

• bounds (tuple[Value, Value]) –

  boundaries depicting transition between the aforementioned regimes, such that:

  • Re <= bounds[0] is considered laminar.

  • bounds[0] < Re <= bounds[1] is considered interim.

  • bounds[1] < Re is considered turbulent.

Returns:

Numpy masks – Laminar, Interim, Turbulent

Return type:

tuple[np.ndarray, np.ndarray, np.ndarray]

Examples

>>> a, b, c = flow_regimes(np.arange(5), (2, 3))
>>> a
array([ True,  True,  True, False, False])
>>> b
array([False, False, False,  True, False])
>>> c
array([False, False, False, False,  True])