17 



APPENDIX 

 EFFECT OF UNDISSOLVED GAS 



The presence of undissolved gas bubbles in either the inflow or outflow can be shown 

 to be detrimental to the accuracy of indication. It is convenient to express the concentration 

 of undissolved gas in terms of the equilibrium pressure which would correspond to its concen- 

 tration if it were dissolved. Let the equilibrium pressure corresponding to the undissolved 

 gas in the inflow be denoted by P^ and that corresponding to the total gas in the inflow be P, 

 so that (P - P ) is the equilibrium pressure corresponding to the dissolved gas in the inflow. 

 Also let P, l^e the equilibrium pressure corresponding to the undissolved gas in the outflow. 

 Then the rate of change of the pressure p in the gas space may be determined as follows: 



It may be assumed that the undissolved gas in the inflow is wholly released into the 

 gas space regardless of the mixing efficiency and the existing pressure, whereas the dissolved 

 portion is brought toward equilibrium with the existing pressure p at a rate dependent upon the 

 mixing efficiency.* The rate of change of the pressure p ascribable to the difference between 

 the concentration of the dissolved gas in the inflow and in the outflow is then given by an 

 expression similar to that appearing in Equation [1], namely, 



^IL{P -P,-p) 



whereas the contribution to the rate at which p changes due to the undissolved gas is 



^(P, - P2) 

 Adding these two contributions gives 



^^Q^(P-P p)^m(p^-p^) [6] 



at V y •■ -^ 



The equilibrium condition dp/dt = requires that r]{P - P. - p) + (Pj - P2) = or 



P. 



p = P+(l-l\p 2. [7] 



Since the total gas content corresponds to a pressure P, the terms on the right side of 

 [7] involving P and P represent the errors arising from undissolved gas in the inflow and in 

 the outflow respectively. It may be observed that while the error due to undissolved gas in 

 the inflow vanishes as the mixing efficiency approaches unity, that due to the same condition 

 in the outflow does not. It is desirable, therefore, to make the mixing efficiency as large as 

 practical and, in addition, to eliminate the convection of significant amounts of undissolved 



*This statement is really a definition of "undissolved gas" combined with the assumption that the gas in the 

 inflow, however, actually dispersed, may be considered to consist of two phases only, dissolved gas and undis- 

 solved gas, each of which behaves as indicated in the above treatment. 



