Medical and Biological Engineering and Computing

, Volume 15, Issue 5, pp 500–512 | Cite as

Analysis of a tubular haemodialyser—effect of ultrafiltration and dialysate concentration

  • R. Jagannathan
  • U. R. Shettigar
Article

Abstract

A theoretical analysis has been made of mass transfer in a hollow-fibre artificial kidney with ultrafiltration for: (i) a constant dialysate concentration and (ii) a variable dialysate concentration along the tube length. The latter is a conjugated boundary-value problem. The solution is obtained by the method of separation of variables using an infinite series expansion considering the fluid to be dilute and Newtonian. Results indicate that the ultrafiltration rate, permeability of the membrane and the nonzero dialysate concentration all have a significant effect on the clearance of the solute. The tubular diameter and length have very little effect on the clearance of the solute for a constant membrane surface area, ultrafiltration rate and a low membrane permeability. At high ultrafiltration rates, the effect of the solute concentration in the dialysate phase on the clearance become negligible. This analysis is compared with those of Popovich et al. (1971), Ross (1974) and Cooneyet al. (1974).

Keywords

Artificial organ Boundary-value problems Mass transfer 

Notation

Notation

A

membrane area

C

concentration;C′=C/C i ;C ++=C′−k

\(\bar C^ +\)

mixing-cup concentration,\((\bar C - C_i )/C_i\)

C+

dimensionless concentration, (C−C i )/C i

Ck

clearance

D

diffusivity

h

mass-transfer coefficient,h o overall value,h c convection, andh d diffusion coefficients

h1(0)

1+Re p /18+83Re p 2/5400

k

(C D /C Bo )/(1+v w (T R −1)/P)

k1

1+v w (T R −1)/P

k2

1+v w T R /P

L

length of the tube

m

mass-transfer rate

NT

number of tubes,N Sho =h o R/D 1,N Shc =h c R/D 1,N Shd =h d R/D 1

P

permeability

Pe

permeation Peclet number,Pe 1=v w R/D 1;\(Pe_2 = (\upsilon _w R/D_2 )(\bar u_{XD} /\bar u(0))\)

ΔP

pressure drop

p

Laplaceian parameter

Q

blood flow rate

q

α m 2 /4δ

R

radius of tube

Rep

permeation Reynolds number,v w R

Shw

wall Sherwood number, (PR/D 1)(1+(v w /P)(T R −1))Sh w1 =PR/D 1,Sh w2 =PR/D 2

s

reduced radius,r/R

TR

transmittance factor

UB

dimensionless axial velocity in the blood phase,\(u_B /\bar u(0)\)

u

axial velocity,u B axial velocity in the blood phase,u xD axial velocity of the dialysate phase,\(\bar u_{xD}\) averageu xD :u yD transverse velocity component in the dialysate phase

\(\bar u(0)\)

average inlet velocity of blood phase

VB

dimensionless radial velocity,v B /v w

vB

radial velocity in the blood phase

vw

wall permeation velocity

X

dimensionless axial length,\(X = (x/R)(\upsilon _w /\bar u(0))\),X L =X atx=L

x

axial length

Y

y/R

y

transverse distance from the wall in the dialysate phase

Z

reduced axial length,xD/u max R 2

λ, α

eigen value

δ

υ w h 1(0)/\(\bar u(0)\)

γ

kinematic viscosity

μ

viscosity

Subscripts

B

blood phase

D

dialysate phase

i

inlet

O

outlet

W

wall

1

blood phase

2

dialysate phase

+

dimensionless

average, Laplace transformed variable

max

maximum value

Sommaire

Une analyse théorique a été entreprise concernant le transfert massif dans un rein artificiel creux en fibre à ultra-filtrage pour (i) une concentration dialytique constante et (ii) une concentration variable sur la longueur du tube. Cette dernière concentration représente un problème de valeur limite conjuguée. La solution est réalisée par la méthode de la séparation des facteurs variables en utilisant une expansion en série infinie et en considérant le fluide comme étant dilué et Newtonien. Les résultats indiquent que le régime ultra-filtrant, la perméabilité de la membrane et la concentration dialytique de non-zéro, ont tous un effet significatif sur l'évacuation du soluté. Par contre, les diamètre et longueur du tube n'ont que très peu d'effet sur l'évacuation du soluté pour une surface de membrane et régime d'ultra-filtrage constants et une faible perméabilité de la membrane. A des régimes d'ultrafiltration élevés, l'effet de la concentration du soluté dans la phase dialytique de l'évacuation, devient absolument négligeable. Cette analyse est ensuite comparée à celles dePopovich et al. (1971), deRoss (1974) et deCooney et al. (1974).

Zusammenfassung

Über die Schlauchlänge wurde eine theoretische Analyse des Stoffaustausches in der künstlichen Hohlfaser-Niere bei Ultrafiltrierung mit (i) konstanter Dialysatkonzentration und (ii) variabler Dialysatkonzentration vorgenommen. Bei der letzteren handelt es sich um ein konjugiertes Grenzwertproblem. Die Lösung wird durch Trennung der Variablen unter Benutzung einer unendlichen Serienexpansion erzielt, bei der die Flüssigkeit als verdünnt und Newtonsch angesehen wird. Die Ergebnisse lassen darauf schließen, daß Filtrierrate, Durchlässigkeit der Membrane und die nicht null betragende Dyalysatkonzentration alle bedeutenden Einfluß auf die Entleerung der gelösten Stoffe haben. Schlauchdurchmesser und-länge haben sehr wenig Einfluß auf den Durchsatz der gelösten Stoffe bei einer konstanten Membranenoberfläche, Ultrafiltrierrate und niedriger Durchlässigkeit der Membrane. Bei hohen Ultrafiltrierraten wird die Auswirkung der gelösten Stoffekonzentration in der Dialysatphase auf den Durchsatz vernachlässigbar. Diese Analyse wird mit Analysen vonPopovich et al. (1971),Ross (1974) undCooney et al. (1974) verglichen.

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Copyright information

© International Federation for Medical & Biological Engineering 1977

Authors and Affiliations

  • R. Jagannathan
    • 1
  • U. R. Shettigar
    • 1
  1. 1.Department of Chemical EngineeringIndian Institute of TechnologyMadrasIndia

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