---
title: "Darcy-Weisbach Equations"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Darcy-Weisbach Equations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(hf)
```




## Introduction

The Darcy-Weisbach equation is the universal and most theoretically rigorous formula for calculating friction head loss in pipe flow. Unlike empirical formulas like Hazen-Williams, Darcy-Weisbach applies to **any fluid** and **any flow regime** (laminar, transitional, or turbulent).

In the `hf` package, the equation is expressed using the volumetric flow rate ($Q$) for consistency:

$$h_f = f \frac{8 L Q^2}{\pi^2 g D^5}$$

Where:

* $h_f$: Friction head loss (m)
* $f$: Darcy friction factor (dimensionless)
* $L$: Pipe length (m)
* $Q$: Volumetric flow rate ($m^3/s$)
* $D$: Internal diameter (m)
* $g$: Acceleration due to gravity ($m/s^2$)



## 1. The Friction Factor ($f$)

The friction factor depends on the Reynolds number and the relative roughness of the pipe. The `hf` package provides two distinct functions to calculate it:

* `calc_friction_cw()`: Uses the implicit **Colebrook-White** equation (default). It finds the exact root iteratively.
* `calc_friction_sj()`: Uses the explicit **Swamee-Jain** equation, a highly accurate approximation that runs faster on massive datasets.

```{r friction_factors}
# Flow parameters
Re <- 100000        # Reynolds number
e <- 0.00026        # Absolute roughness (m)
D <- 0.1            # Diameter (m)

# Compare both methods
calc_friction_cw(reynolds = Re, roughness = e, diameter = D)
calc_friction_sj(reynolds = Re, roughness = e, diameter = D)
```

## 2. Calculating Head Loss with Functional Injection

The `calc_head_loss_darcy` function calculates the head loss, but you can *inject* the friction function you want to use via the `friction_fun` argument.

```{r head_loss}
# 1. Default approach (Uses Colebrook-White automatically)
calc_head_loss_darcy(
  length = 100, flow = 0.02, diameter = 0.1, roughness = 0.00026
)

# 2. Functional Injection (Using Swamee-Jain)
calc_head_loss_darcy(
  length = 100, flow = 0.02, diameter = 0.1, roughness = 0.00026,
  friction_fun = calc_friction_sj
)

# 3. Direct Value (If you already know the friction factor)
calc_head_loss_darcy(
  length = 100, flow = 0.02, diameter = 0.1, friction_factor = 0.025
)
```

## 3. Calculating Diameter and Flow Rate

Because the friction factor creates a circular dependency for diameter and flow rate, the `hf` package uses internal numerical solvers (`uniroot`) to find the exact values. Functional injection works here as well.

```{r iterative_solvers}
# Calculate the required diameter for a target head loss of 8.56m
calc_diameter_darcy(
  loss = 8.56, length = 100, flow = 0.02, roughness = 0.00026
)

# Calculate the maximum flow rate for the same head loss
calc_flow_darcy(
  loss = 8.56, length = 100, diameter = 0.1, roughness = 0.00026
)
```