As already mentioned, the original Colebrook equation is valid for turbulent flow of air, water, or natural gas through pipes. On the other hand, for laminar flow,

${\lambda}_{L\left(p\right)}=64/R{e}_{\left(p\right)}$ is used whereas the transition from laminar to turbulent flow is around

$R{e}_{\left(p\right)}\approx 2320$. This transitional border at the Moody’s plot [

4] is sharp where the equivalent sharp transition from laminar to turbulent flow for the observed fuel cell starts at around

$R{e}_{\left(FC\right)}\approx 500$, as explained in [

8]. Therefore, for airflow through the cathode side of the observed fuel cells, the flow friction factor

${\lambda}_{L\left(FC\right)}$ consists of two clearly defined types of flow:

#### 2.1. Turbulent Flow

In case of turbulent airflow during experiments with open-cathode PEMFCs, measurements show that pressure drop during turbulent flow at its cathode side follows logarithmic law, which form is comparable to the Colebrook’s flow friction equation for pipe flow, but with different numerical values [

8]. The flow friction related to air flow is given by Equation (3):

where:

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless) – the same definition as for pipes

$lo{g}_{10}$—logarithmic function with base 10

$FC$—index related to Fuel Cells

During turbulent flow, numerical values for the flow friction factor in pipe and fuel cells are different and that difference can go up to 60%. To make a direct connection between Equation (1) for pipe flow and Equation (3) for the observed fuel cell, Equation (4) can be used [

25]:

where:

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

${\epsilon}_{\left(FC\right)}$—virtual relative roughness of fuel cell (dimensionless)

$lo{g}_{10}$—logarithmic function with base 10

$FC$—index related to Fuel Cells

For Equation (4), virtual roughness can be recalculated based on the Colebrook equation as

${\epsilon}_{\left(FC\right)}=0.03086$ for the observed fuel cell in the experiment [

8]. This fuel cell was tested with three different cathode configurations [

8]. As noted in [

31], this roughness

${\epsilon}_{\left(FC\right)}$ is not a real measurable physical characteristic of the surface of the used material for conduits (on the contrary for pipes

${\epsilon}_{\left(p\right)}$ can be measured or at least estimated accurately [

32,

33,

34,

35,

36,

37]).

Both, Equations (3) and (4) are numerically unstable for $R{e}_{\left(FC\right)}<575$, which can be a critical problem knowing that turbulent zone starts for $R{e}_{\left(FC\right)}>500$. However, the novel solution proposed in this article is numerically stable.

Generally, implicitly given equations can be transformed in explicit form through the Lambert W-function [

38,

39]. The Lambert W-function [

5] is defined as the multivalued function W that satisfies

$z={e}^{W\left(z\right)}\xb7W\left(z\right)$. However, such transformation for the Colebrook equation for pipes contains a fast-growing term

${e}^{x}$ and because of that, overflow error in computers is possible [

40,

41]. Happily, results with fuel cells show that the solution is not in the zone where

${e}^{x}$ is too big to be stored in registers of computers. The model for fuel cells is given in Equation (5), while the related model for pipe flow friction model can be seen in [

42].

where:

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

${\epsilon}_{\left(FC\right)}$—virtual relative roughness of fuel cell (dimensionless)

${a}_{\left(FC\right)}$, ${b}_{\left(FC\right)}$, ${c}_{\left(FC\right)}$—constants

${x}_{\left(FC\right)}$—variable

$lo{g}_{10}$—logarithmic function with base 10

$ln$—natural logarithm

$e$—exponential function

$W$—Lambert function

$FC$—index related to Fuel Cells

The parameter ${c}_{\left(FC\right)}$ for fuel cell is ${c}_{\left(FC\right)}=0.1415596$.

After procedures from [

6,

7,

43], the following form for fuel cells expressed through the Lambert W-function and its cognate Wright ω-function is given in Equation (6):

where:

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

${a}_{\left(FC\right)}$—constant

${x}_{\left(FC\right)}$, ${\Delta}_{\left(FC\right)}$, ${B}_{\left(FC\right)}$—variable

$ln$—natural logarithm

$W$—Lambert function

$\omega $—Wright function

$FC$—index related to Fuel Cells

However, symbolic regression applied on the explicit formulation, Equation (6), which involves

$W\left({e}^{{x}_{\left(p\right)}}\right)-{x}_{\left(p\right)}=\omega \left({x}_{\left(p\right)}\right)-{x}_{\left(p\right)}$ gives very simple, but still accurate results in case of pipe flow [

6,

7] ([

44,

45] confirm these results), but unfortunately these analytical formulas, which are optimized for pipes, cannot be directly applied on the fuel cell equation. Fortunately, symbolic regression gives also very promising results for fuel cells as given in Equation (7):

where:

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

${x}_{\left(FC\right)}$—variable

$ln$—natural logarithm

$W$—Lambert function

$\omega $—Wright function

$FC$—index related to Fuel Cells

To avoid repetitive computations, parameters

${\Delta}_{\left(FC\right)}$ and

${B}_{\left(FC\right)}$ are introduced, in Equation (8). Both symbolic regression analyses were performed in Eureqa, a commercial software tool, which automates the process of model building and interpretation [

46,

47].

where:

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

${\Delta}_{\left(FC\right)}$, ${B}_{\left(FC\right)}$—variable

$ln$—natural logarithm

$FC$—index related to Fuel Cells

#### 2.2. Unified Model

Although the expression for laminar flow through pipes is

${\lambda}_{L\left(p\right)}=64/R{e}_{\left(p\right)}$, for fuel cells it is different, as given in Equation (9) [

8]:

where:

${\lambda}_{L\left(FC\right)}$—laminar Darcy flow friction factor for fuel cells (dimensionless)

$\frac{h}{H}$—channel depth/channel width used only in laminar flow (dimensionless)

$e$—exponential function

$FC$—index related to Fuel Cells

Values of $h/H$ are from 0.83 to 2.5.

The experiment [

8] shows that air flow through the cathode side of air-forced open-cathode PEMFCs are (1) laminar for the lower values of the Reynolds number,

$R{e}_{\left(FC\right)}<500$ and (2) turbulent for the higher values, 500 <

$R{e}_{\left(FC\right)}$ < 4000, where the Reynolds number is in hydraulics a very well-known dimensionless parameter that is used as a criterion for foreseeing flow patterns in a fluid’s behavior (defined in the same way for air flow through pipes and here discussed air flow through fuel cells). The dimensionless Darcy’s unified flow friction factor

${\lambda}_{\left(FC\right)}$, is the function of the switching function

$y$, the laminar flow friction

${\lambda}_{L\left(FC\right)}$, and the turbulent flow friction

${\lambda}_{T\left(FC\right)}$. The unified coherent flow friction model that covers both laminar and turbulent zones is set by Equation (10) [

30]:

where:

${\lambda}_{\left(FC\right)}$—unified Darcy flow friction factor for fuel cells (dimensionless)

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

${\lambda}_{L\left(FC\right)}$—laminar Darcy flow friction factor for fuel cells (dimensionless)

$y$—switching function

$FC$—index related to Fuel Cells

The novel switching function

$y$ is given in Equation (11):

where:

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

$y$—switching function

$e$—exponential function

$FC$—index related to Fuel Cells

The switching function was obtained by symbolic regression using HeuristicLab [

47] and it is given in

Figure 1.

The laminar flow friction ${\lambda}_{L\left(FC\right)}$ depends on the Reynolds number $R{e}_{\left(FC\right)}$, but also on geometry of conduits, while the turbulent flow friction ${\lambda}_{T\left(FC\right)}$ depends only on the Reynolds number $R{e}_{\left(FC\right)}$. In the case of fuel cells, both coefficients are empirical. In addition, the switching function $y$ contains the exponential function, (the similar situation is for calculation of ${\lambda}_{L\left(FC\right)}$ as already explained).

To avoid numerical instability, it is recommended to use the explicit approximation which gives the following unified formula in Equation (12).

where:

${\lambda}_{\left(FC\right)}$—unified Darcy flow friction factor for fuel cells (dimensionless)

${\lambda}_{T\left(FC\right)}$—turbulent Darcy flow friction factor for fuel cells (dimensionless)

${\lambda}_{L\left(FC\right)}$—laminar Darcy flow friction factor for fuel cells (dimensionless)

$R{e}_{\left(FC\right)}$—Reynolds number (dimensionless)—the same definition as for pipes

$\frac{h}{H}$—channel depth/channel width used only in laminar flow (dimensionless)

${x}_{\left(FC\right)}$, ${\Delta}_{\left(FC\right)}$, ${B}_{\left(FC\right)}$—variables

$y$—switching function

$e$—exponential function

$ln$—natural logarithm

$FC$—index related to Fuel Cells

For 2

^{16} = 65536 Sobol Quasi Monte-Carlo pairs [

48], which cover

$R{e}_{\mathrm{FC}}$ = 50–4100 and for

$h/H$ from 0.83 to 2.45, the maximal relative error of the final calculated flow friction factor

${\lambda}_{\mathrm{FC}}$ using Equation (12) is 0.46% compared with the original Equation (2). The accuracy and speed of execution are tested through the code given in the next section.