Diffusion Through a Biological Membrane

Transcription

1 Diffusion Through a Biological Mebrane B. Background Mass transfer is an iportant concept in cheical engineering. Whenever concentration differences are present ass transfer by diffusion occurs. Diffusion is the echanis of ass transfer caused by olecular otion. For exaple ebranes can be used to separate coponents because saller olecules diffuse faster through a porous ebrane than larger ones. Mebrane separation techniques are based on the differences between diffusion rates through porous ebranes. It is used for separation of gases and purification of liquids. Mebrane processes can be classified as follows:. Gas diffusion through a porous solid. Dialysis (separation of solutes fro a solution via diffusion through a seipereable ebrane) 3. Gas pereation in a ebrane. Solute gas dissolves in the ebrane and then diffuses. 4. Reverse ososis, using pressure to drive a solvent across a seipereable ebrane 5. Ultrafiltration, using pressure to drive a solvent across a seipereable ebrane while retaining the desired solute 6. Gel pereation chroatography (GP), also known as size exclusion chroatography (SE), to characterize the olecular weight distribution of polyers. Saller olecules enter pores, creating a longer path length. In this experient we will consider an exaple of diffusion of a salt (solute) in water solution through a porous planer ebrane. The contacting in a syste consisting of two liquid volues of different solute concentrations separated by a regenerated cellulose ebrane as shown by figure one () will be studied. heical &Bioedical Engineering Dept -USF page of 7

2 Figure opartental Model for the Syste The conditions in the second volue ( ) will be onitored by a conductivity probe. The initial concentration of the water (dialysate) in the second volue is zero. A solute rich volue ( ) will be put in contact with our dialysate in the second volue, separated by a planer ebrane with a thickness (δ). It is iportant to note that we are assuing that there are no external ass transfer effects or occurrences of solutes partitioning (i.e. K) across the ebrane. We are able to assue that there are no external ass transfer effects because the second volue is ixed by a stirring bar and that the second volue is in large excess in coparison to that of the first ( << ). The concentrations of solute in each volue ( and ) are unifor. Mass Transfer Modeling A aterial balance on the syste with an initial aount of solute in the first liquid volue ( equation: total total + o ), assuing that there is no solute in the ebrane, gives the following () heical &Bioedical Engineering Dept -USF page of 7

3 The total aount of solute in the syste is a conserved quantity. The initial aount of solute in the second volue ( ) will be zero ( 0 ), we will be using water as a dialysate and sodiu chloride salt as a solute. () The differential expression for the transfer of the solute into the dialysate is given by equation three (eq.3). This equation neglects volue changes due to diffusion. d d (3) dt dt Fick s law for olecular diffusion states that the ass flux for solute A through species B is proportional to the concentration difference ( ) over the change in path length ( x ) by the apparent diffusivity coefficient ( D ). AB ass flux DAB (4) x This allows for us to evaluate the flux ( area and aount of tie as: N A ) of solute A through species B for a given N A DAB (5) δ The apparent diffusivity coefficient is related to the effective diffusion coefficient through a ebrane ( D eff ). The path of apparent diffusion is unobstructed (shorter).the path length for effective diffusion is uch longer, or tortuous, in order for the solute ( ) to traverse a ebrane of the sae distance. This is depicted in figure. However, apparent diffusion is said not to occur through the ebrane. We will be deterining the effective diffusion coefficient the ebrane. heical &Bioedical Engineering Dept -USF page 3 of 7

4 Figure Apparent versus Effective Diffusion This relationship allows for us to apply Fick s Law to diffusion through an area (A) of a ebrane noral to the path of solute diffusion. d dt D A δ ( ) eff Substitution of an expression for the concentration of the first volue ( ) equation derived fro one (eq.) into six (eq.6) gives the differential equation for the diffusion syste (eq.7). (6) d D eff A D A o eff + dt δ δ By defining new variables a (eq. 7a) and b (eq. 7b), (7) Deff A a δ o (7a) We can rewrite equation seven (eq. 7) as follows. d a b dt Deff A b + δ This ordinary differential equation can be integrated by ethods such as separation of variables. The concentration in the dialysate can be solved for by applying known initial conditions and stated iplicitly (eq. 0) or explicitly (eq. 9). (7b) (8) heical &Bioedical Engineering Dept -USF page 4 of 7

5 bt ( e ) (9) ln bt (0) a (0a) b The concentration as tie approaches infinity ( ) will be deterined experientally (eq. ) as the syste reaches equilibriu. It is iportant to note that we are assuing that the concentration of solute is the sae in both volues (i.e. ). Looking at the liits of equation nine (eq. 9) as tie goes to both infinity and zero gives the following inforation about the syste. Li t t Li 0 ( ) ( + ) ( ) at () This is best understood by looking at a graph of the predicted concentration of the dialysate versus tie (figure 3). This figure shows that, for a short initial period, the concentration rises linearly with a slope equal to a. Hence, if the duration of our experient is too short we will only observe this linear increase. () heical &Bioedical Engineering Dept -USF page 5 of 7

6 Figure 3 Dialysate oncentration versus Tie Refer to you re Transport Phenoena text for ore inforation on diffusion (Geankoplis, h 3). Multiple Batch Processes If we wanted to lower the concentration of the solute further, the process can be repeated using a fresh dialysate. The concentration of the dialysate returns to zero in the new batch. The solute of the first volue will be allowed to, again, for an equilibriu partitioning across the ebrane with the fresh dialysate present in the second volue. This leads to occurrences of ultiple steady states in the first volue. A aterial balance on the new syste takes the following for. total + (3),,, It is iportant to ention that the concentration in the first volue never reaches the value of reached by the dialysate. o, total, As before, this equation can be applied to Fick s Law and the dialysate concentration profile can be calculated by the following equation. ( e ) b t,, (4) Figure four (4) shows the concentration of solute in the second volue versus tie as three dialysate baths are used. heical &Bioedical Engineering Dept -USF page 6 of 7

7 Figure 4- Multiple Batches ontinuing in this fashion the following relation can be developed for the addition of any nuber of new dialysate baths (n). total, n, +, ( n ), n n (5) The clearance ( L ) of solute upon each new batch (n) can be calculated fro the average ass reoval rate ( ave,n ) of a particular solute and the initial concentration of solute in the first volue (note: when n ; L ave ). ave, n, n, n (6) tn ave, n L n (7),( n ) heical &Bioedical Engineering Dept -USF page 7 of 7