7/27/2019 Performance equation for counter current flow artificial kidney device

1/4

Muhammad Valiallah

(Rate of Mass of BUN in at )(Rate of Mass of BUN out at )(Rate of Mass of BUNlost within ) This can be represented with symbols as:

| |() (6)

Simplifying the equation, expressing variables in terms of concentration, dividing through by and taking the limit as :

( )(7)

Similarly the mass balance for the dialysate is given by:

| |() (8)

Note: Mass lost by blood is equal to the mass gained by the dialysate.

( )(9)

Subtracting the above mass balance equations yields

(10)

Integrating the above equation from

(11)

Let ( ) (( ) )

(12)

Rearranging to express the equation in terms of()() (() ) (13)

7/27/2019 Performance equation for counter current flow artificial kidney device

2/4

Muhammad Valiallah

Substituting the above equation into the differential equation that describes the solute

concentration on the blood side

(( ) (() ) )(14)

,( )() ( )-(15)

Integrating the above equation from : ,( )() ( )-

(16)

,( ) ( )-

(17)

Not the integral on the LHS of the equation is of the form:

()

Where are constants ,( ) ( )-

(18)

, ( ) ( )-( )

(19)

Let area: , and number of transfer units: ( ) ( )( ) ( ) ( )

(20)

7/27/2019 Performance equation for counter current flow artificial kidney device

3/4

Muhammad Valiallah

( ) ( ) ( ) (21)

Absorbing the negative sign into the natural logarithm and taking the exponential of the above

equation:

( ) ( ) ,( )-(22)

In order to determine the performance coefficient, the concentration terms must first be

expressed in terms of volumetric flow rate (), mass transfer rate of BUN () across thedialysis membrane and dialysance, which are defined below.

(23)

( ) ( ) (24)

The above definitions are rearranged to a useable form which can readily be substituted into the

solute blood side mass balance equation:

(25)

(26)

,( )-

(27)

Upon simplifying the above equation becomes:

,( )-

(28)

[ ] ,( )-

(29)

7/27/2019 Performance equation for counter current flow artificial kidney device

4/4

Muhammad Valiallah

( ) ,( )-(30)

*,( )- + ,( )-(31)

Divid through by yields: *,( )- + ,( )-

(32)

Rearranging:

*,( )- + ,( )- (33)

The performance equation is by definition:

(34)

,( )- ,( )-

(35)

Relating the differential mass balance for the patient to the artificial kidney device:

( )

( )(36)