Experimental Study of Biodegradation of Ethanol and Toluene Vapors

Appendix E. Determination of Parameter Uncertainty
Combining Equations (5-15a) and (5-30a), the expressions of the
parametersδ and K, can be described by Equations (E-1) and (E-2):
max
(0.635 − 0.5 / Yxe
)
δ =
(E-1)
13 1
3
( + 0.726 − )
max
max
7 Y
Y
xb
xe
13 1
K = ( −1.274)
⋅δ
(E-2)
max
7
Y xb
We assume that the most probable values for δ is given by Equation (E-3):
δ = f Y xe
, Y )
(E-3)
(
xb
The uncertainty in the resulting value forδ can be found by considering the
spread of the values ofδ resulting from combining the individual measurements Y e,i , Y b,i ,
… into individual results of δ i
. The approximation for the standard deviation
δ can be expressed as (Bevington, 1969):
2 2 ∂δ
2 2 ∂δ
2 ∂δ
∂δ
σ δ
≅ σ
Y
( ) + σ
Y
( ) + 2σ
Y Y
( )( )
(E-4)
xe
xb
xe xb
∂Y
∂Y
∂Y
∂Y
xe
xb
2 2 ∂K
2 2 ∂K
2 ∂K
∂K
σ
K
≅ σ
Y
( ) + σ
Y
( ) + 2σ
Y Y
( )( )
(E-5)
xe
xb
xe xb
∂Y
∂Y
∂Y
∂Y
xe
The variance and covariance were obtained as:
xb
xe
xe
xb
xb
σ
δ
for
∂δ
∂
Y xe
∂δ
∂
Y xb
= 29.98
= −0.204
(E-6)
(E-7)
σ = 0.001
(E-9)
Y xe
σ = 0.01
(E-10)
Y xb
190