John S. Cooper
        CSci, CChem, CWEM, FCIWEM, MRSC, DipWEM

    Consultant on Water, Wastewater
    & Environmental Management                                

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John S. Cooper - Consultancy Services

USEFUL FORMULAE IN WATER/WASTEWATER TREATMENT
AND TRADE EFFLUENT DISPOSAL


As an aide memoire the following mathematical formulae are some of those which we have found useful
in dealing with water and wastewater treatment and trade effluent disposal.


Volume of Cylinder

V  =  p r2h

- where  r  =  radius of cylinder
                h  =  height/length of cylinder

Volume of Cone

V  =  1/3p r2h

- where r = radius of base of cone
              h  =  height of cone

Volume of Pyramid

V  =  1/3Ah

- where  A  =  area of base
                h  =  height of pyramid

Volume of Sphere

V  =  4/3 p r3

- where r = radius of sphere

Area of Triangle

(i)    A  =  1/2Lh

- where   L  =  length of base
                 h  =  vertical height

(ii)    A  =  [s(s-a)(s-b)(s-c)]0.5

- where a, b, c = length of sides
                        s = 1/2(a + b + c)

(iii)   A   =   (ab/2)SinC
             =   (ac/2)SinB
             =   (bc/2)SinA

-  where    a, b, c  =   length of sides
                  A , B, C  =   included angles

Area of Sector of Circle

A  =  (p r2/360)a

-  where    r   =   radius of circle
                  a   =   included angle

Area of Segment of Circle

A  =  (p r2/360)a - (c/2)(r - s)

-  where    r  =  radius of circle
                  a  =  included angle
                   c  =  length of chord
                   s  =  depth of segment

Discharge from An Orifice

Q  =  mA(2gH)0.5

-  where   Q  =  discharge (m/s)
                  A  =  area of the orifice (m2)
                  H  =  head over orifice (m)
                  g  =  9.806 (m/s2)
                  m =  ~ 0.62 (sharp-edged plate)
                           ~ 0.62 (penstock)
                           ~ 0.86 (opening in wall of
                                        chamber)
                           ~ 0.81 (short length of pipe)

Flow over 90 Degree V-Notch Weir

Q = 1.4H2.5 (approximate)

-  where   Q  =  flow (m3/s)
                  H  =  head over apex (m)

Flow over Half 90 Degree V-Notch Weir

Q  =  0.7H2.5 (approximate)

-  where   Q  =  flow (m3/s)
                  H  =  head over apex (m)

Flow over a Rectangular Weir
(Fully Contracted)

Q = 1.84(L - 3.9H)H1.5 (approximate)

- where    Q  =  flow (m3/s)
                   L  =  length of weir (m)
                   B  =  head over weir (m)

Flow through a Rectangular Standing
Wave Flume

Q = 1.706BH1.5 (approximate)

- where    Q  =  flow (m3/s)
                   B  =  width of throat (m)
                   H  =  head over invert (m)

Manning Formula
(For Flow in Pipes)

V = (0.01/n)r0.667s0.5

- where     V  =  velocity of flow (m/s)
                    n  =  coefficient of friction
                    r  =  hydraulic mean depth (mm)
                    s  =  hydraulic gradient

Box Pipe Flow Formula
(For Flow in Horizontal Pipes)

Q = 1/5(d5H/L)0.5

- where    Q  =  flowrate (l/s)
                  d  =  diameter (cm)
                  H  =  head (m)
                   L  =  length (m)

Parameter Loading Calculation

kg   =  Conc (mg/l) x Volume (m3)
                               1000

Mogden Formula

General formula frequently employed by the Water
Service Companies for the calculation of trade effluent charges.

C  =  R + V + B(Ot/Os) + S(St/Ss)

- where    C  =  Charge per m3 trade effluent.
                  R  =  Cost per m3 of receiving and conveying
                            sewage through the sewerage system.
                  V  =  Cost per m3 of providing and operating
                            volumetric capacity at the sewage 
                            sewage treatment works.
                  B  =  Cost per m3 of providing and operating
                            biological treatment capacity at the
                            sewage treatment works.
                  S  =  Cost per m3 for the handling and
                           disposal of sludge at the sewage
                            treatment works.
                  Ot  =  Average COD/BOD concentration of
                             trade effluent after settlement for
                            1 hour.
                  Os  =  Average COD/BOD concentration of 
                             sewage after settlement for 1 hour.
                  St  =  Average Settleable Suspended solids
                             concentration of trade effluent.
                  Ss  =  Average Settleable Suspended  Solids
                              concentration of sewage.

Accrued Capital Funds

Total Accrued Capital  =  P[1 + (r/100)]n

- where   P  =  Amount invested initially
                  r   =  Annual rate of interest (%)
                  n  =  No. of years of investment

Annual Capital and Loan Charge Repayments

Annual Repayment   =   Capital Sum                                        [1 - (1 + r)-n]/r

- where    r  =  % Interest Rate/100
                  n  =  Period of loan (years)



©John S. Cooper 2008