attempting to use Newton#nlsolve ... got it, i think. it's fastest when the initial guess is close or the allowed error is huge :)

solve_channel.rb

begin
  require 'bundler/inline'
rescue LoadError => e
  $stderr.puts 'Bundler version 1.10 or later is required. Please update your Bundler'
  raise e
end

gemfile(true) do
  source 'https://rubygems.org'
  gem 'dentaku'
end

require 'bigdecimal/newton'
require 'dentaku'

class Channel

  def initialize
    @flow         = 15
    @z_left       = 3
    @z_right      = 3
    @bottom_width = 3
    @mannings     = 0.024
    @conversion   = 1.486
    @bed_slope    = 0.1

    @zero = BigDecimal::new("0.0")
    @one  = BigDecimal::new("1.0")
    @two  = BigDecimal::new("2.0")
    @ten  = BigDecimal::new("10.0")
    @eps  = BigDecimal::new("1.0e-3")
  end

  def eps; @eps end
  def one; @one end
  def two; @two end
  def ten; @ten end
  def zero; @zero end

  def values(d)
    area = Dentaku("(0.5 * z_left  * d^2) + (0.5 * z_right * d^2) + (bottom_width * d)", {
      z_left:  @z_left,
      z_right: @z_right,
      bottom_width: @bottom_width,
      d: d[0]
    })

    wet_perimeter = Dentaku("((((z_left  * d) ^ 2) + d^2)^0.5) + ((((z_right * d) ^ 2) + d^2)^0.5) + bottom_width", {
      z_left:  @z_left,
      z_right: @z_right,
      bottom_width: @bottom_width,
      d: d[0]
    })

    value = Dentaku("(1/mannings) * conversion * (bed_slope ^ 0.5) * area * (( area / wet_perimiter ) ^ (2/3)) - flow", {
      flow:          @flow,
      mannings:      @mannings,
      conversion:    @conversion,
      bed_slope:     @bed_slope,
      area:          area,
      wet_perimiter: wet_perimeter
    })

    [value]
  end
end


include Newton
c = Channel.new
x = [0.3]
nlsolve(c, x)

p "Channel height is: #{x[0].to_f}"