Pressure Drop

The flow of coolants through the core, pumps, heat exchangers and other coolant cycle components is determined by means of pressure differences. These differences (drops) depend on the cycle itself, and can be divided into several categories.

First and foremost, there are pressure drops due to friction of the coolant with the surrounding walls and due to local changes in flow area, direction and more. These kinds of pressure drops tend to be close to \(\Delta p\propto \rho v^2\), and are usually described by an effective relation following this reasoning:

\[\Delta p = f \frac{\rho u^2}{2}\]

Where \(f\) (sometimes denoted \(K\)) may be dependent non-trivially on geometry, flow properties and regime.

Other contributions come from gravity and inertia.

[1]

Blasius, H., “Das Aehnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten”, Mitteilungen über Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, vol.134, VDI-Verlag Berlin (1913).

[2] (1,2,3)

KAERI, “Development of Research Reactor Technology”, Korea Atomic Energy Research Institute (KAERI), KAERI/RR-3818/2014, 2014.

Friction

Blasius_friction(re)

An early (1913) approximation by Blasius[1] for the turbulent pipe flow Darcy friction factor.

\[f = \frac{0.3164}{\text{Re}^{0.25}}\]

Parameters:

re (Value) – The Reynolds number.

Returns:

f – The Darcy friction factor

Return type:

Value

Examples

>>> Blasius_friction(10_000)
0.03164

Darcy_Weisbach_pressure_by_mdot(mdot, rho, f, L, Dh, A)

The Darcy-Weisbach equation relates pressure loss due to friction in pipe flow to the average velocity of an incompressible fluid:

\[\Delta p = f \frac{\rho u^2}{2}\frac{L}{D_h}\]

In this function, the velocity is replaced by \(\dot{m}=\rho u A\) where u is the average velocity and A the cross-section. The pressure drop is sensitive to the flow’s direction:

\[\Delta p = f \frac{\dot{m}|\dot{m}|}{2\rho A^2}\frac{L}{D_h}\]

If \(\dot{m}>0\) then the pressure returned is positive.

Parameters:

• mdot (KgPerS) – Mass current.

• rho (KgPerM3) – Fluid density.

• f (Value) – The Darcy Friction Factor.

• L (Meter) – Pipe length.

• Dh (Meter) – Hydraulic diameter.

• A (Meter2) – Cross-sectional flow area.

Returns:

dp – Pressure drop across the pipe.

Return type:

Pascal

Examples

>>> Darcy_Weisbach_pressure_by_mdot(mdot=0., rho=1, f=1, L=1, Dh=1, A=1)
0.0
>>> Darcy_Weisbach_pressure_by_mdot(mdot=1., rho=1, f=1, L=1, Dh=1, A=1)
0.5
>>> Darcy_Weisbach_pressure_by_mdot(mdot=-1., rho=1, f=1, L=1, Dh=1, A=1)
-0.5

friction_factor(name, **kwargs)

Create a Darcy friction factor chosen from the list below with uniform signatures. The main usage of this function is as input for pressure_diff.

Available functions:

regime_dependent	regime_dependent_friction, which depends on the Re No., given re_bounds. Importantly, employs viscosity and geometry (e.g. rectangular) corrections upon the laminar and turbulent factors.
laminar	laminar_friction. Does not employ corrections as described above.
turbulent	turbulent_friction
Blasius	Blasius_friction

Parameters:

• name (Literal['regime_dependent', 'laminar', 'turbulent', 'Blasius']) – Method name

• kwargs (Dict) – Options to pass onto the given method

Returns:

f – Darcy friction factor

Return type:

GeneralDarcyFactor

laminar_friction(re)

Due to the Hagen-Poiseuille law, an incompressible Newtonian fluid in laminar flow through a long constant cylindrical pipe yields an analytic Darcy friction factor: \(f=64/\text{Re}\)

Parameters:

re (Value) – The Reynolds number.

Returns:

f – The Darcy friction factor.

Return type:

Value

rectangular_laminar_correction(aspect_ratio)

For a circular pipe, the laminar Darcy friction factor is analytical. For a rectangular pipe, corrections must be made.

In this function such a correction is returned as used in[2].

Parameters:

aspect_ratio (float) – channel_depth / channel_width (less than 1)

Returns:

K_R – geometric correction coefficient

Return type:

float

See also

laminar_friction

Examples

>>> float(rectangular_laminar_correction(1.))
1.1246190353017915
>>> float(rectangular_laminar_correction(0.))
0.6668484123609236
>>> float(rectangular_laminar_correction(0.5))
1.0363896075094456

regime_dependent_friction(T_cool, T_wall, mdot, fluid, pipe, re_bounds, k_R, k_H=None, turbulent=CPUDispatcher(<function turbulent_friction>), laminar=CPUDispatcher(<function laminar_friction>))

A flow-regime-dependent friction coefficient function.

Parameters:

• T_cool (Celsius) – Coolant bulk temperatures

• T_wall (Celsius) – Wall temperatures

• mdot (KgPerS) – Coolant mass flow rate

• fluid (LiquidFuncs) – Coolant properties

• pipe (EffectivePipe) – Pipe geometry

• re_bounds (tuple[float, float]) – Flow regime (Re values) boundaries, see flow_regimes.

• k_R (float) – Geometrical factor. At laminar flow the Darcy coefficient is analytical (see laminar_friction) for a circular duct. k_R is a multiplicative factor such that \(f_\text{laminar} = \frac{64}{\text{Re} k_R}\). See for example, rectangular_laminar_correction. This correction is applied to all regimes.

• k_H (Callable[[float, float], float] | None) – Viscosity factor function, which takes as input \(k_H(P_\text{heat}/P_\text{wet}, \mu_\text{wall} / \mu_\text{bulk})\). By default, it is omitted. See viscosity_correction.

• turbulent (Callable[[Value], Value]) – Strategy for calculating the friction coefficient for turbulent flow.

• laminar (Callable[[Value], Value]) – Strategy for calculation the friction coefficient for laminar flow.

Returns:

f – The Darcy friction factor.

Return type:

Value

turbulent_friction(re, epsilon=0)

An approximation for the friction factor of the implicit Colebrook-White equation, as written in RELAP and[2] page 3, chapter 2.1.2.

For very low Reynolds values (approx. <7), 0.0 is returned.

Parameters:

• re (Value) – Reynolds Number.

• epsilon (Value.) – Relative roughness (roughness height / hydraulic diameter).

Returns:

f – The Darcy friction factor

Return type:

Value

Examples

>>> turbulent_friction(4e3)
0.039804935964641644
>>> turbulent_friction(4e3, 0.1)
0.10560870441248855
>>> turbulent_friction(1e6)
0.011649393290640643
>>> turbulent_friction(5.0)
0.0

viscosity_correction(heat_wet_ratio, mu_ratio)

Temperature differences yield varying viscosity, which affects friction. Thus, a correction must be made.

In this function such a correction is returned as used in[2].

Parameters:

• heat_wet_ratio (Value) – Heated perimeter / Wet perimeter

• mu_ratio (Value) – Viscosity at wall / Viscosity at bulk.

Returns:

K_H – viscosity correction coefficient

Return type:

Value

Examples

>>> viscosity_correction(1., 1.)
1.0
>>> viscosity_correction(1., 0.)
0.0
>>> viscosity_correction(1., 2.)
1.4948492486349383

Local

References

[3] (1,2,3,4,5)

Handbook of Hydraulic Resistance, I. E. Idelchik, ????

bend_factor(angle, relative_curvature, re)

Compute K, the local pressure drop factor for

\[\Delta p = K\rho v^2/2,\]

(where \(\Delta p\) is the pressure drop over a pipe bend, \(\rho\) is the fluid’s density and \(v\) is the fluid’s velocity), according to Idelchik Diagram 6.1[3] (page 424).

This correlation applies to the case of a low curvature smooth pipe bend

\[R_0/D_h < 3\]

(where \(R_0\) is the pipe’s axis radius of curvature, measured from the bend center to the center of the pipe, and \(D_h\) is the hydraulic diameter of the pipe), with circular/square cross-sections and at large Re numbers (above 10,000).

Parameters:

• angle (Radians) – The angle of the bend.

• relative_curvature (float) – The radius of curvature of the pipe’s axis divided by the hydraulic diameter.

• re (float) – The Reynolds number.

Returns:

K – Local pressure drop factor.

Return type:

float

contraction_factor(aratio)

Closed form approximation for sudden contraction at high Re numbers.

According to Idelchik Table 4.10, there is a closed form approximation for sudden contraction of the form \(\frac{1}{2}\left(1-\frac{A_2}{A_1}\right)^{\frac{3}{4}}\). This is written to be true for Re > 35000, but the table for lower Re numbers stops at 10000, so it is unclear what to do for interim values of Re.

Parameters:

aratio (Value) – Area ratio in [0,1]

Return type:

float | ndarray

expansion_factor(aratio)

Compute K, the pressure drop factor (Borda, Carnot), for

\[\Delta p = K\rho v^2/2\]

Where

\[K = \left(1 - \frac{A_1}{A_2}\right)^2\]

According to Idelchik Table 4.2[3], this formula is correct for large Re numbers, above 3300.

Parameters:

aratio (float) – Area ratio in [0,1]

Returns:

K – Pressure drop factor

Return type:

float

Examples

>>> expansion_factor(0. / 1.)
1.0
>>> expansion_factor(1. / np.inf)
1.0
>>> expansion_factor(1. / 2.)
0.25

local_pressure_by_mdot(mdot, rho, f, A)

Calculates the local pressure dot according to a known mass flow rate.

This pressure drop is not the static pressure drop across the local pipe area change, but rather the total pressure drop, i.e. the amount of lost energy in the Bernoulli equation terms.

Following the quadratic velocity relation to local pressure:

\[\Delta p = f \frac{\rho u^2}{2}\]

In this function, the velocity is replaced by \(\dot{m}=\rho u A\) where u is the average velocity and A the cross-section. The pressure drop is sensitive to the flow’s direction:

\[\Delta p = f \frac{\dot{m}|\dot{m}|}{2 \rho A^2}\]

If \(\dot{m}>0\) then the pressure returned is positive.

Parameters:

• mdot (KgPerS) – Mass current.

• rho (KgPerM3) – Fluid density.

• f (Value) – The Darcy Friction Factor.

• A (Meter2) – Cross-sectional flow area.

Returns:

dp – Pressure drop across the pipe.

Return type:

Pascal

Examples

>>> local_pressure_by_mdot(1., 1000., 1., 1.)
0.0005
>>> local_pressure_by_mdot(-1., 1000., 1., 1.)
-0.0005
>>> local_pressure_by_mdot(0., 1000., 1., 1.)
0.0

local_pressure_factor(*, mdot, positive_flow, negative_flow, **kwargs)

Considering \(\dot{m}\), returns K pressure coefficient of positive_flow(A1, A2) or negative_flow(A1, A2).

This pressure coefficient is the loss coefficient for the total pressure in the Bernoulli equation, i.e. actual energy loss, not energy transfer between static and dynamic pressures.

Parameters:

• mdot (KgPerS) – Mass current.

• positive_flow (LocalPressureFactorFunc) – Expansion/Contraction

• negative_flow (LocalPressureFactorFunc) – Expansion/Contraction

Returns:

K – Pressure drop factor

Return type:

float

Examples

>>> def pos(a1, a2): return a1+a2
>>> def neg(a1, a2): return a1*a2
>>> local_pressure_factor(mdot=1., positive_flow=pos, negative_flow=neg, a1=1., a2=2.)
3.0
>>> local_pressure_factor(mdot=-1., positive_flow=pos, negative_flow=neg, a1=1., a2=2.)
2.0

mdot_by_local_pressure(dp, rho, f, A)

The inverse function of local_pressure_by_mdot, which returns \(\dot{m}\) given a known pressure difference.

\[\dot{m} = \text{sign}(\Delta p)\sqrt{\frac{2\rho A^2 |\Delta p|}{f}}\]

If \(\Delta p > 0\) then the current returned is positive.

Parameters:

• dp (Pascal) – Pressure drop across the pipe.

• rho (KgPerM3) – Fluid density.

• f (Value) – The Darcy Friction Factor.

• A (Meter2) – Cross-sectional flow area.

Returns:

mdot – Mass current.

Return type:

KgPerS

Examples

>>> mdot_by_local_pressure(dp=0., rho=1e3, f=1., A=1.)
0.0
>>> mdot_by_local_pressure(dp=1., rho=1., f=2., A=1.)
1.0
>>> mdot_by_local_pressure(dp=-1., rho=1., f=2., A=1.)
-1.0

sudden_contraction_factor(aratio, re)

The Idelchik Table 4.10 interpolation of the local sudden expansion pressure drop coefficient.[3]

The table used is bounded by \(A_2/A_1 \in [0.1, 0.6]\) and \(\text{Re} \in [10, 10000]\). Within those boundaries, linear interpolation is performed.

For larger Reynolds numbers, an analytic solution is used, contraction_factor. For smaller Reynolds numbers, \(K = 50/\text{Re}\) is used.

As for area ratios \(A_2/A_1 \notin [0.1, 0.6]\), where the Reynolds no. is within the table values, an interpolation by the Reynolds no. is performed between the limiting analytic solutions, while for the lower end (\(\text{Re}=10\)) an interpolation by the area ratio is added to match \(K(A_2/A_1=1)=0\) and \(K(A_2/A_1=0)=0.5\) (matching a contraction from an infinite pool,[3]).

The above result may not be applicable for such small Re numbers, but it would not have a large effect since these pressure drops zero out as \(\mathcal{O}(v)\).

Parameters:

• aratio (float) – Area ratio in [0,1]

• re (float) – Reynolds number

sudden_expansion_factor(aratio, re)

The Idelchik Table 4.2 interpolation of the local sudden expansion pressure drop coefficient.[3]

The table used is bounded by \(A_1/A_2 \in [0.1, 0.6]\) and \(\text{Re} \in [10, 3300]\). Within those boundaries, linear interpolation is performed.

For larger Reynolds numbers, an analytic solution is used, expansion_factor. For smaller Reynolds numbers, \(K = 31/\text{Re}\) is used.

As for area ratios \(A_1/A_2 \notin [0.1, 0.6]\), where the Reynolds no. is within the table values, an interpolation by the Reynolds no. is performed between the limiting analytic solutions, while for the lower end (\(\text{Re}=10\)) an interpolation by the area ratio is added to match \(K(A_1/A_2=1)=0\) and \(K(A_1/A_2=0)=1\) (matching an expansion into an infinite pool).

The above result may not be applicable for such small Re numbers, but it would not have a large effect since these pressure drops zero out as \(\mathcal{O}(v)\).

Parameters:

• aratio (float) – Area ratio in [0,1]

• re (float) – Reynolds number

Other Pressure Drop Components

Gravitational Pressure

gravity_pressure(rho, dh, g=9.80665)

In the case of an incompressible barotropic fluid at rest, the force exerted by gravity over a fluid column is \(\Delta p=\rho g\Delta h\)

Parameters:

• rho (KgPerM3) – Fluid density.

• g (MPerS2) – Gravitational acceleration.

• dh (Meter) – Height difference.

Returns:

dp – Pressure drop due to gravity.

Return type:

Pascal

Examples

>>> gravity_pressure(1., 1.)
9.80665
>>> gravity_pressure(1000, 0.)
0.0
>>> # What is the pressure head (for water) on the moon?
>>> gravity_pressure(1000, 1, g=(moon := 1.62))
1620.0

See also

Gravity - a corresponding Calculation.

Inertial Pressure

inertia_pressure(mdot2, dl, A)

A change in flow along a streamline incurs a pressure drop. See further explanation in inertia.

\[\Delta p = \frac{dl}{A}\frac{d\dot{m}}{dt}\]

Parameters:

• mdot2 (KgPerS2) – \(\ddot{m}\), time derivative of mass flow

• dl (Meter) – Length element along the streamline

• A (Meter2) – Area through which the liquid flows in dl.

Returns:

dp – Pressure drop (positive).

Return type:

Pascal

See also

Inertia - a corresponding Calculation.

Helper Functions

A ready made Channel-compatible pressure function is provided:

pressure_diff(T, Tw, fluid, mdot, pipe, dz, f=<function _re_friction.<locals>._f>, g=9.80665, mdot2=None, **_)

Returns pressure difference. Positive direction is assumed downward. This function calculates frictional, inertial, and gravitational pressure drops.

Parameters:

• T (Celsius) – Bulk Coolant temperature

• Tw (Celsius) – Wall temperature

• fluid (LiquidFuncs) – Coolant properties

• mdot (KgPerS) – Coolant mass current

• pipe (EffectivePipe) – Channel geometry

• dz (Meter) – Cell sizes in the flow’s direction

• f (GeneralDarcyFactor | None) – Darcy-Weisbach friction factor, default being friction.Blasius_friction.

• g (MPerS2) – Gravitational acceleration.

• mdot2 (KgPerS2 | None) – \(\ddot{m}\), time derivative of mdot

Returns:

dp – Pressure drop at each cell

Return type:

Pascal

See also

friction.Darcy_Weisbach_pressure_by_mdot, gravity_pressure

static_pressure(pressure, mdot, area, density)

Calculates the static pressure from the total (static+dynamic) Bernoulli term.

Parameters:

• pressure (Pascal) – The total pressure (static+dynamic)

• mdot (KgPerS) – The mass flow rate

• area (Meter2) – The cross-section area of the pipe

• density (KgPerM3) – The fluid density

Return type:

float | ndarray

Examples

>>> static_pressure(1., 1., 1., 1.)
0.5
>>> static_pressure(1., 0., 1., 1.)
1.0
>>> static_pressure(1., -1., 1., 1.)
0.5