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/SSN 014/08/4. Journa/ o/Ihe Soriel)' /or Undemaler Techno/ag)'. Vo/. 23. No. 4. pp. /81-186, /999 Technology

Nomograms for Improving Diver Safety ca -u T. G. ANTHONY* and B. J. HORNt oe'e u * DERA AlverslOke, Fort Road. Gosport. Hampshire POI22DU. UK. G) t Departmel1l of Mathematics and Statistics. University of Call1erbury. Christchurch, NZ. •••

Abstract Similarly, schedules require a diver to breathe appropriate gas mixtures; an Divers are increasingly required to perform error in identifying the gas mixture used, and 'safety to life' calculations which, if incorrect, the associated decompression schedule, may may result in a serious incident. The application result in severe (DCI). of an 'old fashioned' manual computational Traditional training presents with technique, the Nomogram, is proposed as a equations for the required calculations and cheap and simple tool for performing safety to teaches mathematical tricks such as 'Magie life calculations. The Nomogram is friendly to triangle, Dalton's T and the Rotating rectangle' non-mathematicians, applicable to the marine to help simplify their use for the non- environment and may significantly reduce the mathematician [2]. Unfortunately, with serious risk of calculation error. Example Nomograms mistakes being made, these techniques do not are illustrated and the advantages for always result in a successful outcome. A improving diver safety presented. significant advance in training being encouraged by the recreational agencies is to eliminate where possible the use of mathematical 1. Introduction formulae and equations and replace them by 'Look Up' tables. 'Look Up' tables are in the Observation of commercial, military and main easy to use, suitable for use in the extremes recreational diver training procedures shows that of a marine environment and eliminate the risk of divers are increasingly required to undertake calculation errors. They have !imitations, 'safety to life' calculations on the physiological however, in that they can only present a aspects of their gases and their dive limited 'calculation' range and require variables profiles [I, 2]. This is particularly apparent in to be at given values. A large number of tables the growing area of '' and other may need to be produced to cover all combina- ''. These calculations may easily tions of an equation's variables to the required be subject to error either by a simple mistake 01' accuracy. the inappropriate mathematical aptitude of the In our modern electronic and microcomputer diver. In addition, calculations often have to be world, an alternative approach to eliminating the performed in an extreme marine environment. problems of mathematical calculations is to · Typical 'safety to Iife' calculations currently embed them within user-friendly pre-pro- being undertaken by divers include the grammed calculators or computer software. This following: is in many ways an ideal giving correct • Conversion of gas percentage to partial pres- and accurate to a given problem, pro- sures (and vice versa), including the determi- viding the computer adage 'GIGO' (Garbage In nation of maximum safe depth. Garbage Out) is avoided. However, this would require all trainees and divers to obtain these • Calculation of Equivalent Air Depth (EAD) devices or have them available for use. The cost far decompression procedures. of this option may be prohibitively expensive to • Determination of safe flow rates and gas some divers, schools and businesses. mixtures for semi-closed circuit re-breathing The principle presented in this paper is that in a diving apparatus. microcoinputer-based world, an 'old fashioned' A simple calculation error in respect of choosing manual computational technique (i.e., the an appropriate gas mixture, gas flow rate or safe Nomogram) may be app!ied successfully to maximum depth for a dive may easily result in a increasingly complex diving requirements and diver losing consciousness underwater from the application of this technique may significantly hypo- or with a fatal outcome. reduce the risk of diving incidents due to calcula- tion errors. The simplicity, cost and durability of © Crown Copyright 1998-DERA. Publishedwith the the system are appropriate to divers who are non- permission of the Controller of Her Britannic mathematicians and takes account of the Majesty's Stationery Office. extremes of their marine environment.

181 T. G. Anthony and B. J. Horn. Nomogramsfor fmproving Diver Safety

Moreover, although the concept has been pro- of is injected into the breathing circuit at a posed here far diving applications, it is likely to be constant mass flow. The percentage of oxygen of use in any scenario where ca1culations have to inspired by a diver using the apparatus is depen- be undertaken by operators with limited math- dent upon the percentage of oxygen in the gas ematical skills andfor in extreme environments. mixture, the rate of gas flow and the diver's work rate (expressed as the volume of oxygen 2. The Nomogram consumed in one minute [\'02])' The relationship between the inspired percentage of oxygen and The Nomogram has been used for engineering the percentage of oxygen in the gas mixture, and other applications for many years and was rate of gas fIow and diver's work rate is repre- particularly applicable before the development sented by the 'counter-lung equation'. of computer technology [3]. A Nomogram is a two-dimensional diagram representing a math- 0/0 (%02 Mix x 0.01 x Flow) - Y02) /0 2 Ins = . ematical equation in which each variable is repre- (Flow - Y02) sented by one or more graduated lines. To perform the 'ca1culation' a furt her line (known x 100 (3) as the index line) is superimposed on the where Nomogram such that it intersects with each of

the graduated lines at the required value of the %02 Ins percentage of oxygen in the breath- variable. (Examples of calculations using ing circuit inspired by the diver (%) Nomograms are given in Figures 1, 3 and 4 in %02 Mix percentage of oxygen in gas mixture the next section.) entering the breathing circuit (coun- The graduated lines which form the ter-Iung) (%) Nomogram can be straight or curved' depending Flow gas mixture flow rate (litres per min- I on the equation and the method of construction. ute [Imin- ] at Standard Nomograms may be classified by the number of and Dry variables involved: [STPD]) 1 Second class Nomogram represents two vari- Y02 divers oxygen consumption (1min- ables STPD) x = y2 (1) It can be seen from this that to a non-mathe- Third class Nomogram represents three variables matician the 'counter-Iung equation' may be quite daunting and that there is significant scope for x = y2 +2 (2) error in its use. As indicated earlier, a transposi- When constructing Nomograms, the simplest tion or calculation error using this equation may and first approach may result in scales that are lead to an inappropriate gas mixture being not practical in use. A range of Nomogram for- breathed with dire consequences. mats have been identified to overcome these prob- The 'counter-lung' equation has four variables lems [3]. One particular derivation of a third class and can be represented as a fourth Class Nomogram is the 'Circular Nomogram', which Nomogram as shown in Figure I. Note that the we have found particularly applicable to some percentage of oxygen in the gas mixture is given of the equations used in diving. Examples are pre- on a set of lines, each line representing a different sen ted in the next section. value of V02. An example use of the Nomogram is also included in Figure I. 3. Nomograms for Divi ng Appl ications 3.2 Equivalent Air Depth (EAD)for It is not the intention within this paper to show decompress ion the method of constructing a Nomogram but to When considering dive decompression require- illustrate the principles ofuse and application for ments using oxygen in gas mixtures improving diver safety. Nomograms applicable to other than air a principle known as the several diving applications have been developed Equivalent Air Depth may be applied. This deter- to illustrate the use of the technique and are pre- mines the depth breathing air that would give an sented below. The examples also show how to use equivalent of nitrogen to the different formats to ensure useable and practical maximum depth that the gas mixture is being scales. breathed. Decompression may then be conducted using the equivalent air depth as opposed to the 3.1 Gas control in semi-closed circuit re- actual depth. The calculation for this is repre- breathing diving apparatus sented by Equation 4. Although this may be Simple mechanical semi-closed circuit re- viewed as being simpler than the counter-lung breathing diving apparatus uses agas control equation, providing the user can successfully principle to maintain a safe level of inspired identify, remember or derive the equation, it still oxygen whcrc agas containing a fixeu percentage has the potential for miscalculation.

182 Underwater Technology, Vol. 23 No. 4, 1999

%OXYGEN

IN MIXTURE %OXYGEN I I rM1XTUREI EXAMPLE 1 2 3 V0 lmin-1 (STPD) 1 2 3 ~tm-llSTPOJ 2 20 r i j:20 20 20 A 15.0 litres per minute flow rale 01 ! ! 40 % oxygen in nitrogen gas into the i~~~~ counterlung with a 2.0 Ittre V02 will '" f f ~ ':n~~ give a 31 % oxygen in nitrogen 1/ and mixture within the counter-Iung k'._ 2 'ttre per minute I'" V02 30 30 I 50 t i

50 50 I 40 40 r70 (!) I z ; ::> ...J 50 ci: 50 1: 15.0 Ittres per minute w f10w rate into I- counterlung Z DERA ::> ---45673 9 10 " 12 13 U 17 1! 19 ~ 21 ~ n ~ ~ o 'lOW RATE .lmin-1 (STPD) ü60 60 ~ Z W (!) >- 70 70 oX cf.

80 80

90 90

DERA, · I I· ·I ·I ·I ·I ·I .. • I ·1 •.. 1·•••.•• 1 .. ·1•.• 1.. I •.. ·1 ·1 1.. • 1 · .. 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

FLOW RATE -lmin-1 (STPD) © Crown Copyright 1998 - DERA

Figure 1 Nomogram for the counter-Iung equation.

EAD _ % Nitrogen in As Figure 2 shows, the simplest Nomogram metres- % Nitrogen in air construction may result in scales that are not practical in use. Fortunately, there are several sol- x (Depthmetres + 10.13) - ]0.13 (4) utions to the construction of a Nomogram and a An error in determining a correct EAD may lead range of alternative formats may be considered. to inadequate decompression being performed An alternative construction for Third dass resulting in serious DCI. Nomograms is the 'circular' Nomogram [3]. A Figure 2 shows the EAD eqwition as given by 'Circular' construction for the EAD calculation Equation 4 as a Third dass Nomogram formed is presented in Figure 3. In the circular form, from three straight !ines. The structure of this the scales of all three variables are clear and Nomogram is, however, far from ideal and has easy to use. several shortfalls. The example index line (the dotted line on Figure 2) illustrates that all three 3.3 Dalton's Law of Partial scales do not offer adequate resolution or accu- racy over the full scale range and that the angle of A concept that often causes the greatest math- intersection with the % Nitrogen variable line is ematical difficulty in diving is Dalton's Law of too acute. Changing the size of the scales of one Partial Pressures. The Law states that 'in a mix- or more of the variables may reduce these short- ture of gases, the pressure exerted by one of the falls. However, acceptable scales could not be gases is the same as it would exert if it alone achieved within the physical and practical con- occupied the same volume'. A further difficulty straints of a printed sheeL One solution would occurs in that the total pressure of agas mixture be to use more than one Nomogram, where increases with increasing dive depth. Although each nomogram covered part of one of the vari- mathematically the law may be expressed by a ables scales. For example one sheet could have comparatively simple equation (Equation 5), solutions for depths between 6 and 27 m, and problems often occur with the application and another sheet solutions for 27-54 m. associated transpositions of the equation.

183 T. G. Anthony and B. J. Horn. Nomogramsfor Improvil1g Diver Safety

CI) ~ 54 Q) -E 45 I l- n. w 36 Cl a: 27 « I- Z 18 w ...J 9 ~ :J 0 0 W

I I I I I I 10 15 20 25 30 40

% NITROGEN 0

In Q) 9 >- Ci) 18 E I l- 27 n. w 0 36 w <.9 :J 45 « <.9 54 Figure 2 Third Class, Nomogram to determine Equivalent Air Depths.

% O)(YGEN IN BREATHING MIXTURE 30 25 20 40 35 70 75 BO 60 65

% NITROGEN IN BREATHING MIXTURE

DERA EXAMPLE 12 15 18 21 24 27 30 33 36 39 42 30% oxygen in n.itrog9~ "OX'IGVI"BIlEATHlNG*XTUJI! EaUIVALENT AIR OEPTH lEAD) : metres "~"~_1UIIIi ~

EAD . lBmetres.,·- CERA

GAUGEOEPTH metres With a breathing mrxture of 30 % oxygen and 70 % nitrogen 42 ~: ~:~e:I~~f~r06~~(EAD) far deoompression is 18 m

.~ © Crown Copyright 1998 - DERA

Figl.!f'i' 3 f:irCI.lI"H Nnmngmm In crllclJlrlle ECluivalent Air Depths.

184 Ullderwater Teclmology, Val. 23 No. 4, 1999

0/0 OXYGEN IN BREATHING MIXTURE 75 80 70

EXAMPLE 20

HSE P02 LIMITS

1.4 BAR MAXIMUM 1.5 BAR MAXIMUM SELF· CONTAtNED SURFACE SUPPUED BAEAlKING BREATHIHG APPARATU9 I APPARATUS 30% oxygen ExampleA: 0.2 0.5 1.0 1.5 2.0 \ A 3D % 0lCVgen mixture al \ 30 mettes will gfve an oxygen partial pressure DERA \ HSEPO:IUWTS PARTIAL PRESSURE OF OXYGEN (P0 ) of 1.2bar 2 lt.•••...."",.. U~~ bar -~-I-~ 1.2 bar P02\ 0.5 1.0 1.5 2..D CERA PARTIAL PAESSUAE OF bXVGEN (P~) bllr \

Example S: A 1.2 bar P02 ala depth 0' 30 metres will glv8 a 30% oxygen In nitrogen breathlng mixture 21 GAUGEDEPTH metres

42 © Crown Copyright 1998 - DERA

Figure 4 Circular Nomogram for Dallon's Law and P02 limits.

. % gas then limits may be superimposed. This is illus- Partial pressure of gas = ~ trated in Figure 4 by the superimposition of the two eurrent maximum limits for inspired P0 x 2 during diving operations reeommended by the (5) Health and Safety Exeeutive [4].

A typieal applieation of the Law is in respeet of 3.4 Other Nomograms the relationship between inspired partial pressure The Nomograms presented above have been of oxygen (P02), the pereentage of oxygen in the seleeted to indieate the applieation of the teeh- gas mixture and the depth of the dive. The sig- nique for . Several other eonstrue- nificance of this relationship is in respeet of the tions not presented here have been undertaken risk ofhypoxia (P02 less than 0.2 bar) and central to allow ealculation of other diving-related prob- nervous system oxygen toxieity at inereased lems. These inc1ude determination of safe diving inspired partial pressures of oxygen .. It is feit depths for gas mixtures, safe gas flow for semi- that this relationship is onee again best repre- c10sed eireuit re-breathers, identifieation of sented by a 'eireular' Nomogram (Figure 4). appropriate 'Nitrox' deeompression sehedules An additional advantage ofusing a Nomogram and Unit Pulmonary Toxieity Dose (UPTD) as opposed to direet ealculation is that it may be from exposure to inspired P02 levels greater used to impose restrietions on the range over than 0.5 bar. It is feit that a signifieant number whieh a variable (e.g., P02) may be determined. of 'safety to life' and other ealculations may be These restrietions refleet the safe limits of the vari- performed using this teehnique. able being represented. This may be seen in Figure 4. Agas mixture with an inspired P02 of 4. Advantages and Disadvantages less than 0.2 bar would be eonsidered hypoxie, thus the P02 line is limited to a minimum of There are both advantages and disadvantages to 0.2 bar. Should the user try to work below this virtually all ealculation systems. The suitability of minimum value the Nomogram eould not be each system depends upon the abilities of the user used and would indieate that the situation was and the environment in whieh the ealculation is not eonsidered to be safe. being undertaken. It is believed that the If there are severallimits for a variable, or the Nomogram has signifieant advantages for use in Nomogram covers a range of applieations and is diving operations where the operator may not required to allow ealculation beyond a safe level, have the mathematieal aptitude to undertake

185 T. G. Anthony and B. J. Horn. Nomogramsjor Improving Diver Sajety

formal mathematics and the environment may be scaled lines. As indicated with the Nomogram for unsuitable for appropriate electronic aids or the EAD ca1culation it is necessary for the line involved calculations. seales and intereepts to allow aceeptable accuracy Once the initial construction has been und er- for the calculation being undertaken. The safety taken, Nomographs are cheap and easy to pro- advantage gained in limiting the range of values duce (a simple printed sheet of paper). The only that may be calculated using a Nomogram can be requirement for their use is the printed sheet and a a disadvantage in other eircumstanees and straight edge for which purpose a transparent rule reduees the flexibility of the method. is ideal. They may be produeed as robust lami- nated plastic sheets or card for extreme con- 5. Conclusions ditions. Reproduetion in the simplest form is by photocopying, although this should be und er- Divers are inereasingly required to perform taken with caution as some copiers may distort 'safety to life' ealculations whieh, if incorrect, the image. This may be overcome by using stan- may result in a serious incident. Existing tech- dard 'check' calculations. For diver training, niques for redueing the risk of calculation error Nomograms may be easily and cheaply provided are Iimited and not always suitable for use in an to a11members of the dass under instruction. extreme marine environment or appropriate to When correctly constructed, the ca1culation the mathematical aptitude of many divers. system cannot go wrong, providing the straight The use of an almost forgotten technique, the edge is not a110wed to 'slip'. As the ca1culation Nomogram, is presented as a cheap and simple procedure is quick, this eventuality may be over- tool for performing 'safety to life' calculations. 1t come by repeating the calculation two or three is feit that the Nomogram has significant advan- times. An alternative approach is to draw the tages for use during diving operations. The risk of Index line on a Nomogram thus providing a per- an incorreet ealculation being made is reduced manent record of the calculation performed. For and safety limits may be included. audit purposes this may be taken to an extreme where the index line drawn on the Nomogram is signed, checked, countersigned and filed as a per- References manent record. Concern for diver safety is often based on the I. Royal Navy Diving Manual, BR2806; physiological effects of the diver breathing an CINCFLEETjFSAG/P2808j2, Change 3 March incorreet gas mixture for the conditions of the 1993. dive. The Nomogram enables safety limits to be 2. AODC and The Underwater Centre, 1986, The Diving Supervisors Manual, Ist Edition, Wallace, clearly presented or forced on the diver by either Traverse-Healyand Regester Ltd. induding them on the Nomogram or restricting 3. Alcack and lanes, The Nomogram, 1938,Sir Isaac the variable range which the system covers. Pitman & Sons, Ltd. There are also limitations to the use of 4. HSE projects inland/inshore- Nomograms, the greatest ofwhich is the potential Approved Code 01 Practice, 1998, HMSO, ISBN for reduced accuracy due to the resolution of the 0717614956.

186