Notation for use in the description of wastewater treatment processes

War. Res. Vol. 21, No. 2, pp. 135-139, 1987 Printed in Great Britain

0043-1354/87 $3.00+0.00 Pergamon Journals Ltd

REPORT NOTATION FOR USE IN THE DESCRIPTION OF WASTEWATER TREATMENT PROCESSES In 1980 a Working Group was set up by the International Association on Water Pollution Research & Control (IAWPRC) and the Commission on Water Quality of the International Union of Pure and Applied Chemistry (IUPAC) to prepare a proposal for unifying notation used in the description of biological wastewater treatment processes. This action was motivated by the benefits that would result from the adoption of a common system of notation in the dissemination of results in international publications. For this purpose the Working Group reviewed journals and books within the English, French, German and American literature in order to establish the quantities most often symbolized and to determine the symbols most commonly used to denote these quantities. By making use of established practice and common acceptance of recognized symbols it was hoped to gain the support of authors in the adoption of a unified system of notation. The Working Group consists of: P. GRAU (Czechoslovakia), Chairman P. M. SUTTOr~ (United States), Secretary M. H~Nz£ (Denmark) S. ELMALL~I(France) C. P. GRADY (United States) W. GUJER (Switzerland) J. KOLLER(Czechoslovakia). The first report of the Working Group appeared in the following journals: Water Research, Vol. 16, pp. 1499-1505, 1982. Pure and Applied Chemistry, Vol. 55, pp. 1035-1040, 1983. Sciences et Techniques De L'Eau, Vol. 16, pp. 275-278, 1983. T.S.M. L'Eau, Vol. 79, pp. 94-98, 1984. Vodni hospodarstvi, Vol. 33, pp. 209-212, 1983. Indian Assoc. Wat. Poll. Con. Newsletter, 20-01, Jan. 1983. The recommendations of the report were accepted by the International Association on Water Pollution Research & Control and the International Union of Pure and Applied Chemistry. The report was widely circulated to professional societies and interested authors throughout the world in order to obtain the views of all organizations and individuals on the recommended standard system of notation. Since its publication, comments have been received from many authors acknowledging that the system would be of value in preparing their own papers and interpreting the papers of others. Although it was originally intended that a period of 1-yr should be allowed to obtain such views, this period did not allow sufficient time for authors to become familiar with the notation system through actual usage in preparing technical articles and consequently the time frame was extended. Specific changes to the notation were recommended by a number of reviewers. These recommendations were considered collectively by the Working Group and resulted in the revised standard notation system which follows. This revised system is likely to be modified again at some future date to reflect new concepts, processes, and required symbols, as well as new views and opinions of authors. INTRODUCTION The number of quantities for which symbols have been chosen is purposely limited to those commonly used in the description of biological wastewater treatment processes. The symbol chosen for each quantity is based on established practice where possible. The quantities have been tabulated under several different headings, a symbol specified for each, and the dimension for the quantity stated. Dimensions have been denoted as follows: length mass time temperature mole

= = = = = 135

L M T 0 tool.

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Although it is recognized that "0" and "tool" are not dimensions, these symbols have been utilized for convenience. Also, subscripts to dimensions have been utilized. Although unusual, this practice historically has been found useful in biological wastewater treatment notation. The authors must state the specific units used for each quantity within the SI system. For example, pressure which has the dimensions ML -~ T -2, can be expressed in units such as kilopascal, newton per m 2, etc. Definitions have not been stated for the quantities. It is expected that the author will provide a definition for a quantity when requested as per footnote 7. Symbol

Quantity name or names

Dimension

Footnote

L L L L L2 L-~ (from L 2 L -3) L3 (from L 3 L 3) Mi L 3 Mi L-3 MiL 3

1,2 1,2 1,2

I. Geometric dimensions, amounts, concentrations H or h L or 1 6 d A a

Height or depth Length Thickness Diameter Area Specific interfacial area

V

Volume Intergranual porosity or void volume fraction Particulate material concentration Soluble material concentration Total material concentration

E

X S C

II. Thermodynamic parameters T Temperature p Total pressure pp Partial pressure R Ideal gas constant P Power III. Physical properties of matter D Diffusion coefficient, dispersion r/ Dynamic viscosity v Kinematic viscosity ¢r Surface tension p Density IV. Time, velocities, and flow rates t Chronological or running time 0 Space time (volume of reaction phase divided by volumetric flow rate of phase entering the reactor) f Mean hydraulic residence time Ox Mean solids residence time u Velocity v Superficial velocity g G Q QG D

Gravitational acceleration Velocity gradient Liquid volumetric flow rate Gas volumetric flow rate Dilution rate

R

Recycle ratio

0 ML-IT 2 ML J T l ML2T -2 0 -j mol -~ ML 2T -3 L 2 T -I

ML I T - I L 2T-I MT-2 ML -3 T T T T LT -) LT~I (from L 3 L -2 T -l) LT-2 T-I L3T-I L3T-I T-I

5 6 3,7 8

3,5 4,7

(from L 3L -3 T -1)

V. Parameters for mass flow rate, mass loading rate, and heat and mass transfer F

NA J BA

Mass flow rate Suspended solids loading rate per unit area (such as clarifier loading rate) Mass flux or solids flux Mass flow rate per unit area

Mi T - 1 MiL -2 T -

1,3,7

Mi L-2 T-1 M~L -2 T -

1,3,7

1

1

Report Symbol By

Bx k K

/3

Quantity name or names Mass flow rate per unit volume or volumetric loading rate, Raumbelastung, etc. Mass loading rate per unit mass, Schlammbelastung, etc. Mass transfer coefficient Overall mass transfer coefficient Oxygen transfer ratio Oxygen saturation ratio

VI. Reaction rates and stoichiometry Reaction rate per unit area rA rv Reaction rate per unit volume rx Reaction rate per unit mass q Reaction rate per unit biomass t~ or q ~ Maximum reaction rate per unit biomass /z Specific biomass growth rate /i or #m~

Maximum specific biomass growth rate

gob*

Observed or net specific biomass growth rate Specific biomass loss rate

Y

ro~, P

Biomass yield coefficient Observed or net biomass yield coefficient Stoichiometric coefficient Process rate

VII. Parameters for reaction rates and conversions n Reaction order k or K Reaction rate coefficient or constant, saturation constant, half velocity coefficient, etc. /(i Inhibition constant E

Ao /¢

E

f t/

Activation energy Pre-exponential factor or frequency factor Temperature coefficient Efficiency Fraction (such as fraction of active organisms, etc.) Effectiveness factor (for example, correction of reaction rate for diffusion)

137 Dimension

Footnote

Mi L-3 T -1

1, 3, 7

MiM~-1T -1 LT -I LT -1

1, 3, 6, 7 9 9

--

7

--

7

MiL-2 T -l MiL-3 T -1 MiM~"l T -1 MiM~-I T -1 MiM~-J T -1 T -l (from MxM~-j T -1) T -l (from MxM~-1 T -j T -1 (from MxM~-1 T -I) T -j (from MxM~-l T -l) MxMi-1 MxMi-1 MiMj-1 Dimension depends on definition

1, 3,10 1,3, 10 1, 3, 10, 11 I, l l 1,3, II 1

Dimension depends on kinetic model Dimension depends on kinetic model ML 2 T -2 mol -j Dimension depends on kinetic model 0 -1

4, 10, 14

--

1,3 1,3 1, 12 1 1, 3 1 4, 13

3 15 16 17 4

4

FOOTNOTES 1. The subscripts i and j refer to the compounds or materials as defined by the author. Subscript X specifically refers to the nature of the solids as defined by the author (e.g. suspended solids, volatile suspended solids). See notes under "subscripts" for further explanation. 2. Particulate material concentration plus soluble material concentration equals the total material concentration (C = X + S). If material is known to be soluble (e.g. oxygen, nitrate, nitrite), the authors should use the symbol S. 3. The subscript used here is an inherent part of the symbol. For further exlanation see notes under "subscripts". 4. Use this symbol provided no confusion is possible with same symbol being used for another quantity. If confusion is possible, add subscript to symbol for representation of one of the quantities. Such a subscript becomes an inherent part of symbol. An example is the use of D for dilution rate and Ds for dispersion coefficient. 5. F o r gases, temperature and pressure conditions must be stated. 6. The value of/" must be calculated from residence time distribution function.

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7. The authors must supply a definition, preferably by equation. 8. This quantity and the representative symbol applies to hydraulic loading rate to biological reactors (trickling filters, rotating biological contactors, fluidized beds, etc.), overflow rates from clarifiers, etc. When specified for biological reactors it represents empty bed velocity. 9. Appropriate subscripts must be used with k and K as mass transfer coefficients (e.g. kc and kL). The subscripts are an inherent part of the symbol. 10. The use of dS/dt, dC/dt, dX/dt, as a general definition of reaction rate (r) is valid for batch reactors only. The appropriate equation for r is derived from a material balance around the reactor. 11. The symbols q, t~, or qmaxshould be used for reaction rate per unit biomass vs r+ for reaction rate per unit mass in general (see example illustrating use of notation). In using symbols, sign convention must be specified (example; removal positive sign, production negative sign). 12. The symbol b may refer to mechanisms such as decay, endogenous metabolism, death, predation, etc. which may be distinguished by subscripts as defined by the author. Such subscripts become an inherent part of the symbol. 13. Process rate is always positive and is defined for example by r,/v in which case it would have the dimension MiL-3T -l. 14.The type of reaction or order or reaction may be specified by a subscript if necessary. 15. Activation energy as used in the equation: k = A0 e x p ( - E / R T ) . 16. Pre-exponential factor as used in the equation: k = A0 e x p ( - E / R T ) Dimension of A0 dependent on dimension of reaction rate coefficient (k). 17. Temperature coefficient as used in the equation:

This equation is preferred over:

kr~ = exp[x( TI - T2)]. k~

kr I = k~0(71

72}

(0 being a temperature coefficient) which is dimensionally not homogeneous. SUBSCRIPTS The information contained in the subscripts is organized in the following manner and order or sequence: 1. Subscripts as an inherent part of the symbol as specified in symbol list (e.g. 0x and #max). Such subscripts must always be used. 2. Alphabetical subscripts representing the type of compound or material. Such subscripts are only necessary if the author is symbolizing more than one type of compound or material. The abbreviations used as subscripts should be defined by the author and may be based on the language in which the paper is written. Examples: SBOD= SBSa = BV,TBOD = BV,TBSB=

soluble BOD5 concentration Konzentration von BSB 5 in Losung total BOD5 volumetric loading rate Raumbelastung mit totalem BSB 5.

3. Numerical or lower case alphabetical subscripts identifying the location to which the symbol applies. Such subscripts are only necessary if symbols are being used to represent compounds or materials at different locations in a flow sheet. The location to which the symbol applies must be defined by the author by means of a flow diagram or a list. 4. Subscripts specifying miscellaneous information such as temperature, statistical properties, etc. Examples: krj = reaction rate at temperature T1 YM= mean biomass yield coefficient. Note: The subscripts belonging to different groups (1--4) should be separated by a comma. SPECIAL NOTATION 1. For dimensionless variables use the overscript (~). Example: 0" = dimensionless space time. 2. Time should be indicated by use of brackets following the symbol and on the same line. Examples:

C(t) = total material concentration at time t C(5) = total material concentration at time equal to 5 min (or h, etc.)

Report

139

2. No superscripts (asterisks, primes, etc.) are allowed. ILLUSTRATING THE USE OF STANDARD NOTATION To illustrate the use of standard notation for biological wastewater treatment, two examples are presented of equations referenced from the literature and then translated into a form reflecting the use of the proposed standard symbols. In addition, a process flow sheet using numerical subscripts to identify the location to which the symbol applies is presented. TRANSLATION OF REFERENCED EQUATIONS Referenced form

Acceptable translated form

k YS

K,+S

~YS -b I~ob~= K + S

b

where: net specific biomass growth rate biomass yield coefficient maximum specific rate of substrate utilization concentration of substrate microorganism decay rate K,= half-velocity coefficient

Y= k-~. S= b=

~obs =Y

=, =S =b --K

Lawrence A. W. and McCarty P. L. (1970) Unified basis for biological treatment design and operation. J. sanit. Engng Div., Am. Soc. cir. Engrs 96 (SA3), 757-778. So 1 + kaXvt

$1 1 + kXvO

S = - -

$2=--

where: S = So = ka = Xv= t =

effluent substrate concentration influent substrate concentration substrate removal rate coefficient mixed liquor volatile suspended solids retention time in aeration basin

= = = = =

$2 S~ k Xv 0

Eckenfelder W. W. Jr (1970) Water Quality Engineering For Practicing Engineers, 162 pp. Barnes & Noble, New York. Process flow sheet Anoxic reactor

Aerobic reactor

where:

influent flow rate influent total BOD5 concentration v,= anoxic reactor liquid volume v~= aerobic reactor liquid volume Xvss, 2 -- aerobic reactor volatile suspended solids concentration Q3 = recircuiation flow rate Q6 = waste sludge flow rate Q7 = return sludge flow rate SBOD,2 ---~soluble BOD5 concentration in aerobic reactor etc. QO

CBOD, 0

Settling