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Overview & Outline Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 5a Chapter 5b Chapter 5c Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Appendix A Appendix B Appendix C Appendix D

CHAPTER 12 — Mixed-Gas Diving Theory 12-1

CHAPTER 12

0L[HG*DV'LYLQJ7 KHR U\

12-1

INTRODUCTION

12-1.1

Purpose.

The fundamental laws and concepts of underwater physics presented in

Chapter 2 (Volume 1) are basic to a proper understanding of mixed-gas diving

techniques. In mixed-gas diving, calculations requiring the use of the various gas

laws are vital to safe diving. A thorough working knowledge of the application of

the gas laws is mandatory for the mixed-gas diver. This chapter reviews the gas

laws.

12-1.2

Scope.

This chapter discusses the theory and techniques used in mixed-gas

diving.

12-2

BOYLE’S LAW

Boyle’s law states that at constant temperature, the absolute pressure and the

volume of gas are inversely proportional. As pressure increases, the gas volume is

reduced; as the pressure is reduced, the gas volume increases.

The formula for expressing Boyle’s law is:

Where:

C is constant

P is absolute pressure

V is volume

Boyle’s law can also be expressed as:

Where:

P

1

= initial pressure

V

1

= initial volume

P

2

= final pressure

V

2

= final volume

When working with Boyle’s law, absolute pressure may be measured in atmo-

spheres absolute. To calculate absolute pressure using atmospheres absolute:

or

CPV

×

=

P

1

V

1

P

2

V

2

=

P

ata

Depth fsw 33 fsw+

33 fsw

---------------------------------------------------

=

P

ata

psig 14.7 psi+

14.7 psi

-------------------------------------

=

12-2 U.S. Navy Diving Manual—Volume 3

Sample Problem 1.

The average gas flow requirements of a diver using a MK 21

MOD 1 UBA doing moderate work is 1.4 acfm when measured at the depth of the

diver. Determine the gas requirement, expressed in volume per minute at surface

conditions, for a diver working at 132 fsw.

1.

Rearrange the formula for Boyle’s law to find the initial volume (V

1

):

2.

Calculate the final pressure (P

2

):

3.

Substitute known values to find the initial volume (V

1

):

4.

The gas requirement for a diver working at 132 fsw is 7.0 acfm.

Sample Problem 2.

Determine the gas requirement, expressed in volume per

minute at surface conditions, for a diver working at 231 fsw.

1.

Rearrange the formula for Boyle’s law to find the initial volume (V

1

):

2.

Calculate the final pressure (P

2

):

3.

Substitute the known values to find the initial volume (V

1

):

The gas requirement for a diver working at 231 fsw is 11.2 surface acfm.

V

1

P

2

V

2

P

1

---------------

=

P

2

132 fsw 33 fsw+

33 fsw

--------------------------------------------

=

5 ata=

V

1

5 ata 1.4 acfm

×

1 ata

------------------------------------------

=

7.0 acfm=

V

1

P

2

V

2

P

1

-------------

=

P

2

231 fsw 33 fsw+

33 fsw

--------------------------------------------

=

8 ata=

V

1

8 ata 1.4 acfm

×

1 ata

------------------------------------------

=

11.2 acfm=

CHAPTER 12 — Mixed-Gas Diving Theory 12-3

Sample Problem 3.

Determine the gas requirement, expressed in volume per

minute at surface conditions, for a diver working at 297 fsw.

1.

Rearrange the formula for Boyle’s law to find the initial volume (V

1

):

2.

Calculate the final pressure (P

2

):

3.

Substitute the known values to find the initial volume (V

1

):

The gas requirement for a diver working at 297 fsw is 14.0 surface acfm.

Sample Problem 4.

An open diving bell of 100-cubic-foot internal volume is to

be used to support a diver at 198 fsw. Determine the pressure and total surface

equivalent volume of the helium-oxygen gas that must be in the bell to balance the

ambient water pressure at depth.

1.

Calculate final pressure (P

2

):

2.

Rearrange the formula to solve for the initial volume (V

1

):

3.

Substitute the known values to find the initial volume (V

1

):

V

1

P

2

V

2

P

1

-------------

=

P

2

297 fsw 33 fsw+

33 fsw

--------------------------------------------

=

10 ata=

V

1

10 ata 1.4 acfm

×

1 ata

---------------------------------------------

=

14.0 acfm=

P

2

198 fsw 33 fsw+

33 fsw

--------------------------------------------

=

7 ata=

V

1

P

2

V

2

P

1

---------------

=

V

1

7 ata 100 ft

3

×

1 ata

-------------------------------------

=

700 ft

3

=

12-4 U.S. Navy Diving Manual—Volume 3

There must be 700 ft

3

of helium-oxygen gas in the bell to balance the water pres-

sure at depth.

Sample Problem 5.

The open bell described in Sample Problem 4 is lowered to

297 fsw after pressurization to 198 fsw and no more gas is added. Determine the

gas volume in the bell at 297 fsw.

1.

Calculate the final pressure (P

2

):

2.

Rearrange the formula to solve for the final volume (V

2

):

3.

Substitute the known values to find the final volume (V

2

):

The gas volume in the bell at 297 fsw is 70 ft

3

.

12-3

CHARLES’/GAY-LUSSAC’S LAW

Charles’ and Gay-Lussac’s laws state that at a constant pressure, the volume of a

gas is directly proportional to the change in the absolute temperature. If the pres-

sure is kept constant and the absolute temperature is doubled, the volume will

double. If temperature decreases, volume decreases. If volume instead of pressure

is kept constant (i.e., heating gas in a rigid container), then the absolute pressure

will change in proportion to the absolute temperature.

The formula for expressing Charles’/Gay-Lussac’s law when the pressure is

constant is:

Where:

V

1

= initial volume

V

2

= final volume

T

1

= initial absolute temperature

T

2

= final absolute temperature

P

2

297 fsw 33 fsw+

33 fsw

--------------------------------------------

=

10 ata=

V

2

P

1

V

1

P

2

-------------

=

V

2

7 ata 100 ft

3

×

10 ata

-------------------------------------

=

70 ft

3

=

V

2

V

1

T

2

T

1

-------------

=

CHAPTER 12 — Mixed-Gas Diving Theory 12-5

The formula for expressing Charles’/Gay-Lussac’s law when the volume is

constant is:

Where:

P

1

= initial absolute pressure

P

2

= final absolute pressure

T

1

= initial absolute temperature

T

2

= final absolute temperature

Sample Problem 1.

The on-board gas supply of a PTC is charged on deck to 3,000

psig at an ambient temperature of 32°C. The capsule is deployed to a depth of 850

fsw where the water temperature is 7°C. Determine the pressure in the gas supply

at the new temperature. Note that in this example the volume is constant; only

pressure and temperature change.

1.

Transpose the formula for Charles’/Gay-Lussac’s law to solve for the final

pressure:

2.

Convert Celsius temperatures to absolute temperature values (Kelvin):

°K = °C + 273

T

1

= 32°C + 273 = 305°K

T

2

= 7°C + 273 = 280ºK

3.

Convert initial pressure to absolute pressure:

4.

Substitute known values to find the final pressure:

P

2

P

1

T

2

T

1

------------

=

P

2

P

1

T

2

T

1

------------

=

P

1

3 000 psig

,

14.7 psi+

14.7 psi

-------------------------------------------------------

=

205 ata=

P

2

205 ata 280

°

K

×

305

°

K

-----------------------------------------

=

188.19 ata=

12-6 U.S. Navy Diving Manual—Volume 3

5.

Convert the final pressure to gauge pressure:

The pressure in the gas supply at the new temperature is 2749 psig.

Sample Problem 2.

A habitat is deployed to a depth of 627 fsw at which the water

temperature is 40°F. It is pressurized from the surface to bottom pressure, and

because of the heat of compression, the internal temperature rises to 110°F. The

entrance hatch is opened at depth and the divers begin their work routine. During

the next few hours, the habitat atmosphere cools down to the surrounding sea

water temperature because of a malfunction in the internal heating system.

Determine the percentage of the internal volume that would be flooded by sea

water assuming no additional gas was added to the habitat. Note that in this

example pressure is constant; only volume and temperature change.

1.

Convert Fahrenheit temperatures to absolute temperature values (Rankine):

°R = °F + 460

T

1

= 110°F + 460

= 570°R

T

2

= 40°F + 460

= 500°R

2.

Substitute known values to solve for the final volume:

3.

Change the value to a percentage:

V

2

= (0.88

×

100%) V

1

= 88% V

1

4.

Calculate the flooded volume:

Flooded volume = 100% - 88%

= 12%

P

2

188.19 ata 1 ata–

()

14.7 psi

()×

=

2 751.79 psig

,

=

V

2

V

1

T

2

T

1

-------------

=

V

1

500

°

R

570

°

R

----------------

×

=

0.88V

1

=

CHAPTER 12 — Mixed-Gas Diving Theory 12-7

Sample Problem 3.

A 6-cubic-foot flask is charged to 3000 psig and the

temperature in the flask room is 72°F. A fire in an adjoining space causes the

temperature in the flask room to reach 170°F. What will happen to the pressure in

the flask?

1.

Convert gauge pressure unit to absolute pressure unit:

2.

Convert Fahrenheit temperatures to absolute temperatures (Rankine):

3.

Transpose the formula for Charles’s/Gay-Lussac’s law to solve for the final

pressure (P

2

):

4.

Substitute known values and solve for the final pressure (P

2

):

The pressure in the flask increased from 3,000 psig to 3,570.03 psia. Note that

the pressure increased even though the flask’s volume and the volume of the

gas remained the same.

12-4

THE GENERAL GAS LAW

The general gas law is a combination of Boyle’s law, Charles’ law, and Gay-

Lussac’s law, and is used to predict the behavior of a given quantity of gas when

pressure, volume, or temperature changes.

P

1

3 000 psig

,

14.7+=

3 014.7 psia

,

=

°

R

°

F 460+=

T

1

72

°

F460+=

532

°

R=

T

2

170

°

F 460+=

630

°

R=

P

2

P

1

T

2

T

1

------------

=

P

2

3 014.7 psia

,

630

°

R

×

532

°

R

-----------------------------------------------------

=

1 899 261

,,

532

°

R

-------------------------

=

3 570.03 psia

,

=

12-8 U.S. Navy Diving Manual—Volume 3

The formula for expressing the general gas law is:

Where:

P

1

= initial absolute pressure

V

1

= initial volume

T

1

= initial absolute temperature

P

2

= final absolute pressure

V

2

= final volume

T

2

= final absolute temperature

The following points should be noted when using the general gas law:

There can be only one unknown value.

If it is known that a value remains unchanged (such as the volume of a tank) or

that the change in one of the variables will be of little consequence, cancel the

value out of both sides of the equation to simplify the computations.

Sample Problem 1.

A bank of cylinders having an internal volume of 20 cubic

feet is to be charged with helium and oxygen to a final pressure of 2,200 psig to

provide mixed gas for a dive. The cylinders are rapidly charged from a large

premixed supply, and the gas temperature in the cylinders rises to 160°F by the

time final pressure is reached. The temperature in the cylinder bank compartment

is 75°F. Determine the final cylinder pressure when the gas has cooled.

1.

Simplify the equation by eliminating the variables that will not change. The

volume of the tank will not change, so V

1

and V

2

can be eliminated from the

formula in this problem:

2.

Multiply each side of the equation by T

2

, then rearrange the equation to solve

for the final pressure (P

2

):

3.

Calculate the initial pressure by converting the gauge pressure unit to the

atmospheric pressure unit:

P

1

V

1

T

1

-------------

P

2

V

2

T

2

-------------

=

P

1

T

1

------

P

2

T

2

------

=

P

2

P

1

T

2

T

1

------------

=

P

1

2 200 psig

,

14.7 psi+=

2 214.7 psia

,

=

CHAPTER 12 — Mixed-Gas Diving Theory 12-9

4.

Convert Fahrenheit temperatures to absolute temperature values (Rankine):

5.

Fill in known values to find the final pressure (P

2

):

6.

Convert final pressure (P

2

) to gauge pressure:

The pressure when the cylinder cools will be 1896.3 psig.

Sample Problem 2.

Using the same scenario as in Sample Problem 1, determine

the volume of gas at standard temperature and pressure (STP = 70°F @ 14.7 psia)

resulting from rapid charging.

1.

Rearrange the formula to solve for the final volume (V

2

):

2.

Convert Fahrenheit temperatures to absolute temperature values (Rankine):

°R = °F + 460

T

1

= 160°F + 460

= 620°R

T

2

= 70°F + 460

= 530°R

°

R

°

F 460+=

T

1

160

°

F 460+=

620

°

R=

T

2

75

°

F460+=

535

°

R=

P

2

2 214.7 psia

,

535

°

R

×

620

°

R

------------------------------------------------------

=

1 911.07 psia

,

=

P

2

1 191.07 psig

,

=

1 896.3 psig

,

=

V

2

P

1

V

1

T

2

P

2

T

1

-------------------

=

12-10 U.S. Navy Diving Manual—Volume 3

3.

Fill in known values to find the final volume (V

2

):

Sample Problem 3.

Determine the volume of the gas at STP resulting from slow

charging (maintaining 70°F temperature to 2,200 psig).

1.

Rearrange the formula to solve for the final volume (V

2

):

2.

Convert Fahrenheit temperatures to absolute temperature values (Rankine):

3.

Substitute known values to find the final volume (V

2

):

Sample Problem 4.

A 100-cubic-foot salvage bag is to be used to lift a 3,200-

pound torpedo from the sea floor at a depth of 231 fsw. An air compressor with a

suction of 120 cfm at 60°F and a discharge temperature of 140°F is to be used to

inflate the bag. Water temperature at depth is 55°F. To calculate the amount of time

required before the torpedo starts to rise (neglecting torpedo displacement,

breakout forces, compressor efficiency and the weight of the salvage bag), the

displacement of the bag required to lift the torpedo is computed as follows:

1.

Calculate the final volume (V

2

):

V

2

2 214.7 psia

,

20ft

3

×

530

°

R

×

14.7 psia 620

°

R

×

--------------------------------------------------------------------------

=

2 575.79 ft

,

3

STP=

V

2

P

1

V

1

T

2

P

2

T

1

-------------------

=

T

1

75

°

F460+=

535

°

R=

T

2

70

°

F460+=

530

°

R=

V

2

2 214.7 psia

,

20ft

3

×

530

°

R

×

14.7 psia 535

°

R

×

--------------------------------------------------------------------------

=

2 985.03 ft

,

3

STP=

V

2

3200 lbs

64 lb / ft

3

-------------------------

=

50ft

3

=

CHAPTER 12 — Mixed-Gas Diving Theory 12-11

2.

Calculate the final pressure (P

2

):

3.

Convert Fahrenheit temperatures to absolute temperature values (Rankine):

4.

Rearrange the formula to solve for the initial volume (V

1

):

5.

Substitute known values to find the initial volume (V

1

):

6.

Compute the time:

(Note that the 140°F compressor discharge temperature is an intermediate

temperature and does not enter into the problem.)

12-5

DALTON’S LAW

Dalton’s law states that the total pressure exerted by a mixture of gases is equal to

the sum of the pressures of the different gases making up the mixture, with each

gas acting as if it alone occupied the total volume. The pressure contributed by any

gas in the mixture is proportional to the number of molecules of that gas in the

P

2

231 fsw 33 fsw+

33 fsw

--------------------------------------------

=

8 ata=

°

R

°

F 460+=

T

1

60

°

F460+=

520

°

R=

T

2

55

°

F460+=

515

°

R=

V

1

P

2

V

2

×

T

1

×

P

1

T

2

×

-------------------------------

=

V

1

8 ata 50 ft

3

×

520

°

R

×

1 ata 515

°

R

×

--------------------------------------------------------

=

403.8 ft

3

=

Time

Volume Required

Compressor Displacement

-----------------------------------------------------------------------

=

403.8 ft

3

120 ft

3

/ min

---------------------------------

=

:03::22=

12-12 U.S. Navy Diving Manual—Volume 3

total volume. The pressure of that gas is called its partial pressure (pp), meaning

its part of the whole.

The formula for expressing Dalton’s law is:

Where: A, B, and C are gases and

Sample Problem 1.

A helium-oxygen mixture is to be prepared which will

provide an oxygen partial pressure of 1.2 ata at a depth of 231 fsw. Compute the

oxygen percentage in the mix.

1.

Convert depth to pressure in atmospheres absolute:

2.

Calculate the oxygen percentage of the mix.

Since:

Then:

The oxygen percentage of the mix is 15 percent.

Sample Problem 2.

A 30-minute bottom time dive is to be conducted at 264 fsw.

The maximum safe oxygen partial pressure for a dive under normal operating

conditions is 1.3 ata (Table 14-4). Two premixed supplies of HeO

2

are available:

84/16 percent and 86/14 percent. Which of these mixtures is safe for the intended

dive?

P

Total

pp

A

pp

B

pp

C

…

+++=

pp

A

P

Total

%Vol

A

×

100%

---------------------------------------

=

P

Total

231 fsw 33 fsw+

33 fsw

--------------------------------------------

=

8 ata=

pp

A

P

Total

%Vol

A

100%

------------------

×

=

%Vol

A

pp

A

P

Total

--------------

100%

×

=

1.2 ata

8 ata

-----------------

100%

×

=

15% oxygen=

CHAPTER 12 — Mixed-Gas Diving Theory 12-13

1.

Convert depth to pressure in atmospheres absolute:

2.

Calculate the maximum allowable O

2

percentage:

Result: The 14 percent O

2

mix is safe to use; the 16 percent O

2

mix is unsafe.

1.26 ata O

2

is less than the maximum allowable.

Use of this mixture will result in a greater risk of oxygen toxicity.

Sample Problem 3.

Gas cylinders aboard a PTC are to be charged with an HeO

2

mixture. The mixture should provide a ppO

2

of 0.9 ata to the diver using a MK 21

MOD 0 helmet at a saturation depth of 660 fsw. Determine the oxygen percentage

in the charging gas, then compute the oxygen partial pressure of the breathing gas

if the diver makes an excursion from saturation depth to 726 fsw.

1.

Convert depth to pressure in atmospheres absolute:

P

Total

264 fsw 33 fsw+

33 fsw

--------------------------------------------

=

9 ata=

%Vol

A

pp

A

P

Total

--------------

100%

×

=

1.3 ata

9 ata

-----------------

100%

×

=

14.4% oxygen=

The pp of the 14% mix 9 ata

14%

100%

--------------

×

=

1.26 ataO

2

=

The pp of the 16% mix 9 ata

16%

100%

--------------

×

=

1.44 ataO

2

=

P

Total

660 fsw 33 fsw+

33 fsw

--------------------------------------------

=

21 ata=

12-14 U.S. Navy Diving Manual—Volume 3

2.

Calculate the O

2

content of the charging mix:

3.

Convert excursion depth to pressure in atmospheres absolute:

4.

Calculate the O

2

partial pressure at excursion depth:

12-6

HENRY’S LAW

Henry’s law states that the amount of gas that will dissolve in a liquid at a given

temperature is almost directly proportional to the partial pressure of that gas. If

one unit of gas is dissolved at one atmosphere partial pressure, then two units will

be dissolved at two atmospheres, and so on.

%VolO

2

0.9 ata

21 ata

-----------------

100%

×

=

4.3% O

2

=

P

Total

726 fsw 33 fsw+

33 fsw

--------------------------------------------

=

23 ata=

p

pO

2

23 ata

4.3% O

2

100%

---------------------

×

=

0.99 ata=