Applies the Hardy-Cross method to solve for pipe flows in a network.

This function uses the Hardy-Cross method to iteratively solve the equations for conservation of mass and energy in a water pipe network. The input consists of a data frame with the pipe characteristics and lists of the pipes in each loop (listed in a clockwise direction) and the initial guesses of flows in each pipe (positive flows are in a clockwise direction).

Usage

hardycross(
  dfpipes = dfpipes,
  loops = loops,
  Qs = Qs,
  n_iter = 1,
  units = c("SI", "Eng"),
  ret_units = FALSE
)

Arguments

dfpipes: data frame with the pipe data. Format is described above, but must contain a column named _ID_.
loops: integer list defining pipes in each loop of the network.
Qs: numeric list of initial flows for each pipe in each loop [$m^3 s^{-1}$ or $ft^3 s^{-1}$]
n_iter: integer identifying the number of iterations to perform.
units: character vector that contains the system of units [options are SI for International System of Units and Eng for English (US customary) units.
ret_units: If set to TRUE the value(s) returned for pipe flows are of class units with units attached to the value. [Default is FALSE]

Value

Returns a list of two data frames:

dfloops - the final flow magnitude and direction (clockwise positive) for each loop and pipe
dfpipes - the input pipe data frame, with additional columns including final Q

Details

The input data frame with the pipe data must contain a pipe ID column with the pipe numbers used in the loops input list. There are three options for input column of the pipe roughness data frame:

Column Name	Approach for Determining K
ks	f calculated using Colebrook equation, K using Darcy-Weisbach
f	f treated as fixed, K calculated using Darcy-Weisbach
K	K treated as fixed

In the case where absolute pipe roughness, $ks$ (in m or ft), is input, the input pipe data frame must also include columns for the length, $L$ and diameter, $D$, (both in m or ft) so $K$ can be calculated. In this case, a new $f$ and $K$ are calculated at each iteration, the final values of which are included in the output. If input $K$ or $f$ columns are provided, values for $ks$ are ignored. If an input $K$ column is provided, $ks$ and $f$ are ignored. If the Colebrook equation is used to determine $f$, a water temperature of $20^{o}C$ or $68^{o}F$ is used.

The number of iterations to perform may be specified with the n_iter input value, but execution stops if the average flow adjustment becomes smaller than 1 percent of the average flow in all pipes.

The Darcy-Weisbach equation is used to estimate the head loss in each pipe segment, expressed in a condensed form as $h_f = KQ^{2}$ where: $$K = \frac{8fL}{\pi^{2}gD^{5}}$$ If needed, the friction factor $f$ is calculated using the Colebrook equation. The flow adjustment in each loop is calculated at each iteration as: $$\Delta{Q_i} = -\frac{\sum_{j=1}^{p_i} K_{ij}Q_j|Q_j|}{\sum_{j=1}^{p_i} 2K_{ij}Q_j^2}$$ where $i$ is the loop number, $j$ is the pipe number, $p_i$ is the number of pipes in loop $i$ and $\Delta{Q_i}$ is the flow adjustment to be applied to each pipe in loop $i$ for the next iteration.

Examples


#              A----------B --> 0.5m^3/s
#              |\   (4)   |
#              | \        |
#              |  \       |
#              |   \(2)   |
#              |    \     |(5)
#              |(1)  \    |
#              |      \   |
#              |       \  |
#              |        \ |
#              |   (3)   \|
# 0.5m^3/s --> C----------D

#Input pipe characteristics data frame. With K given other columns not needed
dfpipes <- data.frame(
ID = c(1,2,3,4,5),                     #pipe ID
K = c(200,2500,500,800,300)            #resistance used in hf=KQ^2
)
loops <- list(c(1,2,3),c(2,4,5))
Qs <- list(c(0.3,0.1,-0.2),c(-0.1,0.2,-0.3))
hardycross(dfpipes = dfpipes, loops = loops, Qs = Qs, n_iter = 1, units = "SI")
#> Using fixed K values
#> Iteration: 1, Loop: 1, dQ: -0.02805
#> Iteration: 1, Loop: 2, dQ: 0.02000
#> $dfloops
#>   loop pipe        flow
#> 1    1    1  0.27195122
#> 2    1    2  0.05195122
#> 3    1    3 -0.22804878
#> 4    2    2 -0.05195122
#> 5    2    4  0.22000000
#> 6    2    5 -0.28000000
#> 
#> $dfpipes
#>   ID    K           Q
#> 1  1  200  0.27195122
#> 2  2 2500  0.05195122
#> 3  3  500 -0.22804878
#> 4  4  800  0.22000000
#> 5  5  300 -0.28000000
#>