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1993

# Diffusion dialysis of sulfuric and nitric acids

Belzahet Trevino

University of Rhode Island

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Trevino, Belzahet, "Diffusion dialysis of sulfuric and nitric acids" (1993). Open Access Dissertations. Paper 3926.

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# Diffusion dialysis of sulfuric and nitric acids

Treviño, Belzahet, Ph.D.

University of Rhode Island, 1993
<figure><img src="image_1.png" /></figure>
DIFFUSION DIALYSIS OF SULFURIC AND NITRIC ACIDS

BY

BELZAHET TREVIÑO

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
CHEMICAL ENGINEERING

UNIVERSITY OF RHODE ISLAND

1993
ABSTRACT

Diffusion dialysis "DD" is defined as a selective transfer of anions (acid) across a positively charged membrane by diffusion, from a concentrated solution to a dilute solution, in which cations (metals) have been rejected. DD is an alternative to the traditional way of treating spent acid baths (neutralization), providing a means of recycling the separated acid. The transport of ions across a charged membrane can be modeled by the extended form of the Nernst-Planck flux equation, which is affected greatly by the acid and metal concentration, temperature, and hydrostatic pressure difference "ΔhP". It is our objective to provide an understanding of the above factors on the performance of DD, in order to optimize a particular separation.

Sulfuric acid/ferrous sulfate and nitric acid/copper nitrate solutions were chosen as sample spent acid solutions. A continuous countercurrent anion-exchange system was used to perform the DD experiments. Copper in nitric acid has a higher leakage rate across the membrane than iron in sulfuric acid. This difference correlates with the low ion hydrated volume and low dissociation constant for copper. DD of nitric acid is more affected by temperature than sulfuric acid, however, the overall mass transfer coefficient "K" for nitric acid is much higher than the one for sulfuric. The differences in K correlate with the hydrated anion volume. Osmosis increases as temperature increases. The reduction in the recovery volume by osmosis increases the recovered acid normality "%N", but decreases the amount recovered "%Rec".

The K increases with temperature but the increase resistance by osmosis over-powers it, reducing the overall acid flux. Increasing iron concentration in the feed increases
the recovered acid normality "%N", but decreases the amount recovered "%Rec".

The K increases with temperature but the increase resistance by osmosis over-powers it, reducing the overall acid flux. Increasing iron concentration in the feed increases the sulfate ion flux due to Donnan equilibrium. Copper does not increase the nitrate flux due to its high metal leakage and monovalent relationship between nitrate and hydrogen in Donnan equilibrium. Increasing the acid feed concentration reduces the %Rec. This reduction correlates with an increase in the ΔhP. The ΔhP can be adjusted by modifying the recovery outlet level. High ΔhP reduces the %Rec and increases the %N. There is always a compromise between these two parameters. The ability to adjust the recovery outlet level enables the unit to achieve optimum conditions for a particular separation.
ACKNOWLEDGEMENTS

With the completion of this thesis, a dream that started more than 10 years ago turns into a reality that I, my self, some times find very hard to believe. When I finally come to accept that the dream is over and reality is here, I realize that this dream come true was not done solely by my own hands, but the work and support of many different people, that without them my dream could never happened. It is my true intention to give every single person that has supported me the greatest acknowledgement. I'll like to thank CONACYT for all the unlimited financial support that gave me the opportunity to come to the University of Rhode Island and start my Ph.D. program. I'll like to thank the faculty of the Department of Chemical Engineering for giving me the opportunity to be in the graduate program, and obtained a great education from their teaching and advise. I'll like to thank Dr. Stanley M. Barnett for accepting me as his graduate student, introducing me to knew research subjects and finding the financial support to carry me during all my program, to him my unlimited gratitude. I'll like to thank the Department of Environmental Management for accepting me in their pollution prevention program providing the financial support and industrial experience in the pollution prevention area that has helped me enormously. I'll like to thank Mr. Richard Enander for accepting me in his pollution prevention program and give me the opportunity to work with a great group of people. Thank you, Gene Park, Richard Girasole, Joe Haberek, Dave Durang, Terry Fox and Maryelen Cobb, for letting me be part of your team. I'll like to thank my family for supporting me during all this time to pursued my dream. I'll like to thank my mother for giving me the love and reassurance a needed during, my brothers for giving me the help and support I needed, to all of you my love with all my heart. I'll like to thank Marina Castillo for encouraging and helping me during difficult times. Finally I'll like to thank GOD for giving me every single thing I have accomplished.
# PREFACE

This dissertation is written in manuscript form. All the manuscripts have been prepared for the submission to "Journal of Membrane Science". These manuscripts are presented in chapters III, IV, and V in this dissertation.

Chapter I is the introductory paper in which the subject of dialysis is reviewed, covering from the history of dialysis to the transport phenomena involved in the process. Chapter II is a detail materials and methodology of the entire dissertation, including tables of trivial calculations. Chapter III is the effect of temperature on diffusion dialysis performance, including differences between water and feed temperature. Chapter IV is the effect of acid and metal concentration on diffusion dialysis performance, emphasizing the effect of acid concentration on the hydrostatic pressure. Chapter V is the effect of the recovery outlet level on the diffusion dialysis performance, emphasizing it as a tool to control the hydrostatic pressure difference across the membrane.
TABLE OF CONTENTS

<table><thead><tr><th>CHAPTER</th><th>PAGE</th></tr></thead><tbody><tr><td>I. Introduction and Review</td><td>1</td></tr><tr><td>History of Dialysis</td><td>2</td></tr><tr><td>Dialysis Membranes</td><td>3</td></tr><tr><td>Dialysis Systems</td><td>6</td></tr><tr><td>Dialysis Principles & Definition</td><td>7</td></tr><tr><td>Transport Phenomena in Dialysis</td><td>10</td></tr><tr><td>III. Materials and Methods</td><td>35</td></tr><tr><td>Preparation of Required Solutions</td><td>36</td></tr><tr><td>Diffusion Dialysis Equipment</td><td>36</td></tr><tr><td>Diffusion Dialysis Setup</td><td>38</td></tr><tr><td>Analysis of Diffusion Dialysis Performance</td><td>41</td></tr><tr><td>III. Temperature Effect on Diffusion Dialysis Performance</td><td>51</td></tr><tr><td>Summary</td><td>52</td></tr><tr><td>Introduction</td><td>53</td></tr><tr><td>Experimental</td><td>58</td></tr><tr><td>Results and Discussion</td><td>60</td></tr><tr><td>Conclusion</td><td>70</td></tr><tr><td>References</td><td>72</td></tr><tr><td>IV. Acid and Metal Concentration Effect on Diffusion Dialysis Performance</td><td>99</td></tr><tr><td>Summary</td><td>100</td></tr><tr><td>Introduction</td><td>101</td></tr><tr><td>Experimental</td><td>103</td></tr><tr><td>Results and Discussion</td><td>105</td></tr><tr><td>Conclusion</td><td>113</td></tr><tr><td>References</td><td>114</td></tr></tbody></table>
<table><tr><td>V. Hydrostatic Pressure Difference Effect on Diffusion Dialysis Performance</td><td>131</td></tr><tr><td>Summary</td><td>132</td></tr><tr><td>Introduction</td><td>133</td></tr><tr><td>Experimental</td><td>136</td></tr><tr><td>Results and Discussion</td><td>137</td></tr><tr><td>Conclusion</td><td>140</td></tr><tr><td>References</td><td>142</td></tr><tr><td>VI. Bibliography</td><td>153</td></tr></table>
LIST OF TABLES

<table><thead><tr><th>CHAPTER II</th><th>PAGE</th></tr></thead><tbody><tr><td>Table 1. Diffusion Dialysis Run Data Sheet</td><td>44</td></tr><tr><td>Table 2. Volume Calculations Data Sheet</td><td>45</td></tr><tr><td>Table 3. Titration Data Sheet</td><td>46</td></tr><tr><td>Table 4. Atomic Absorption Data Sheet</td><td>47</td></tr><tr><td>Table 5. Overall Mass Transfer Coefficient Data Sheet</td><td>48</td></tr><tr><td>Table 6. Hydrostatic Pressure Data Sheet</td><td>49</td></tr></tbody></table>
# LIST OF FIGURES

## PAGE

## CHAPTER I

### FIGURE

1. Countercurrent Diffusion Dialyzer 34

## CHAPTER II

### FIGURE

1. Diffusion dialysis equipment 47

2. Diffusion dialysis equipment 48

3. Diffusion dialyzer 49

## CHAPTER III

### FIGURE

1. Diffusion dialyzer 74

2. Diffusion dialysis equipment 75

3 & 3A. Effect of temperature on pressure difference across the membrane 76 - 77

4 & 4A. Effect of temperature on recovery and waste volumes 78 - 79

5 & 5A. Effect of temperature on acid recovery and normality percentage 80 - 81

6 & 6A. Activation energy by Arrhenius plot 82 - 83

7 & 7A. Effect of temperature on pressure difference across the membrane 84 - 85

8 & 8A. Effect of temperature on recovery and waste volumes 86 - 87

9 & 9A. Effect of temperature on acid recovery, normality, and metal leakage 88 - 89

10 & 10A. Activation energy by Arrhenius plot 90 - 91

11 & 11A. Effect of water temperature on pressure difference across the membrane 92 - 93

12 & 12A. Effect of water temperature on recovery and waste volumes 94 - 95

13 & 13A. Effect of water temperature on acid recovery, normality, and metal leakage 96 - 97
# CHAPTER IV

## FIGURE

<table><tr><td>1.</td><td>Diffusion dialyzer</td><td>116</td></tr><tr><td>2.</td><td>Diffusion dialysis equipment</td><td>117</td></tr><tr><td>3 & 3A.</td><td>Effect of acid concentration on pressure difference across the membrane</td><td>118-119</td></tr><tr><td>4 & 4A.</td><td>Effect of acid concentration on recovery and waste volumes</td><td>120-121</td></tr><tr><td>5 & 5A.</td><td>Effect of acid concentration on acid recovery, normality, and metal leakage</td><td>122-123</td></tr><tr><td>6 & 6A.</td><td>Effect of metal concentration on the pressure difference across the membrane</td><td>124-125</td></tr><tr><td>7 & 7A.</td><td>Effect of metal concentration on recovery and waste volumes</td><td>126-127</td></tr><tr><td>8 & 8A.</td><td>Effect of metal concentration on acid recovery, normality, and metal leakage</td><td>128-129</td></tr></table>

# CHAPTER V

## FIGURE

<table><tr><td>1.</td><td>Diffusion dialyzer</td><td>144</td></tr><tr><td>2.</td><td>Diffusion dialysis equipment</td><td>145</td></tr><tr><td>3 & 3A.</td><td>Effect of recovery outlet level on recovery and waste hydrostatic pressure</td><td>146-147</td></tr><tr><td>4 & 4A.</td><td>Effect of pressure difference across the membrane on recovery and waste volumes</td><td>148-149</td></tr><tr><td>5 & 5A.</td><td>Effect of pressure difference across the membrane on acid recovery, normality, and metal leakage</td><td>150-151</td></tr></table>
# Chapter I
# INTRODUCTION AND REVIEW

## History of Dialysis

Use of dialysis for commercial separations began at the turn of the twentieth century with the recovery of sodium hydroxide (NaOH) from cellulose steeping liquors in the rayon industry, and the recovery of sulfuric acid ($H_2SO_4$) from copper leaching solutions in the electrolytic copper refining industry. A great deal of effort was expended in an attempt to realize the potential saving of sodium hydroxide and sulfuric acid in their respective industries. Despite the many features of dialysis which were attractive, commercial success was not achieved mainly because of the deterioration of the membranes under acid or alkali media. The first membranes available were cellulosic materials like cellophane and parchmentized kraft paper.

Industry and the process itself required membranes which were stronger, thinner, and resistant to corrosive solutions. The need was fulfilled by the development of ion-selective membranes for saline water reclamation by electrodialysis. Strong, efficient and acid resistant membranes were made out of polyvinyl chloride (PVC) by the Phelps Dodge Copper Company and the Graver Water Conditioning Company in the 50's, later other companies like Tokuyama Soda Company, Ltd. and Asashi Chemical Company, Ltd. followed in developing dialyzers and membranes on a commercial scale (Perry and Green, 1984). Another big factor for the improvement of dialysis membranes, was the development of the artificial kidney (hemodialysis) by Kolff and Berk (1944) during World War II. Hemodialysis systems have been developed to the point where long-term maintenance of life by periodic hemodialysis is practical in
many cases. At this time more than 200,000 end-stage kidney patients are maintained world-wide by hemodialysis (Drukker, Parsons, and Maher, 1983).

## Dialysis Membranes

Initial membranes were designed to obtain solute separation mainly by differences in molecular size. For desalination, ion-exchange membranes were developed giving higher selectivity to dialysis separation.

A polymeric matrix to which ionizable groups are covalently bound swells and imbibes water when immersed in an aqueous solution. Dissociation of the ionizable groups leaves charges bound to the matrix and counterions free in the absorbed water. When prepared in membrane form such polymeric matrices are called ion exchange membranes (Mears, 1983). Ion-exchange membranes can be viewed as a sheet of ion-exchange resin. Cation-exchange membranes are permeable to cations but not anions. Anion-exchange membranes are permeable to anions but not to cations. The source of this ability to discriminate between cations and anions is based on the fixed negative or positive charges on the polymeric chains. Negative charges (cationic membrane) repulse negative ions that try to enter the membrane and exclude them, but positive ions (cations) can. If positive fixed charges (anionic membrane) are attached to the polymer chain instead of negative fixed charges, positive ions cannot permeate the membrane, but negative ions (anions) can (Klein et al, 1987). There are two basic components of an ion-exchange membrane: a support matrix made out of polymer chains, and functional groups provided by carboxylic and sulfonic acid (negative charge) or quaternary amine and pyridinium salts (positive charge).
(Meares, 1983). The polymer matrix gives the mechanical strength and limits the changes in lateral dimensions, it also increases the resistance to water and ion transfer. The type of polymer as well as the crosslinking density may vary according to the type of membrane needed, a common example of an anion-exchange membrane is polystyrene matrix with quaternary ammonium (NR<sup>+3</sup>) groups crosslinked with divinylbenzene. Other membrane materials are: regenerated cellulose, cellulose acetate, ethylene-polyvinyl alcohol, polyacrylonitrile, polycarbonate, polymethylmethacylate, polyperfluoro, polysulfone and polystyrene (Davis, 1992; Klein et al, 1987). Functional groups are attached to the polymer by covalent bounds, the type of charge and the fixed-charge density provide the membrane with the ion selectivity. There are two types of commercially available ion-exchange membranes: heterogeneous and homogeneous membranes. Heterogeneous membranes, in which ion exchange particles are incorporated in film-forming resins by mixing ion-exchange resin powder and film-forming polymer powder into a fabric. Homogeneous ion-exchange membranes have been made by graft polymerization of anionic or cationic moieties into preformed films.

For an ion-exchange membrane to be practical for low-cost processing it must have a low resistance to ion transfer as well as being ion selective. To decrease resistance, a membrane may be made thinner, the cross linking density may be decreased, or fixed charge density may be increased. Decreases in crosslinking density tend to increase the length and flexibility of polymer segments between points of crosslinking. This increases the size of the void spaces between segments, which
makes it easier for ions to move through the membrane. Increases in fixed-charge
density also tend to increase the size of the void spaces because of the repulsion of
like fixed charges. But, if the void spaces between polymer chains become too large
due to any of the above factors, there can be volumes in the center of the voids that
are not affected by the fixed charges on the chains. This results in ineffective
repulsion of the undesired ions and lower selectivity. Usually a compromise between
selectivity and low resistance must be made. Membranes available now, combine
excellent selectivity with low resistance, high physical strength, and long life (Klein et
al, 1987). In addition to the traditional membranes, there are some special ion-
exchange membranes with the following properties: selectively permeable to
monovalent anions or cations, highly crosslinked to reduce diffusion of non-
electrolytes, loosely crosslinked to pass large ions, anion-exchange membranes
permeable to acid diffusion, anion-exchange membranes that retard acid diffusion,
cation-exchange membranes, permeable to base diffusion, resistant to organic fouling
, resistant to organic solvents, oxidation resistant, high pH tolerant, low electrical
resistant, mechanical strength and water-splitting bipolar membranes
(Davis, 1992; Lacey, 1988).

Dialysis Systems and Applications

Dialysis units were developed in three different configurations: (1) The tank type, in which the membrane, supported upon a rigid frame, formed a flat bag unit, to be immersed in a tank. An example of this type is the Cerini dialyzer, developed in Italy in 1928 and used in the rayon industry. This technique has a low surface to
volume ratio and is not suited for continuous operations. (2) The filter-press type, consisted of an assembly of frames with a membrane between each adjoining pair and held together by a clamping device. A recent example is the plate-and-frame design by Graver (Graver, 1993). (3) The tube type, or hollow-fiber module, consisted of a bundle of vertical membrane tubes which achieved commercial importance in hemodialysis applications (Kessler and Klein, 1992; Tuwiner, 1962).

The plate-and-frame type is the most widely used today. In countercurrent operation the flow pattern is arranged so that, as shown in Figure 1, the lower density dialysate (water) flows downward and the higher density feed (spent acid) flows upward in adjacent compartments. Thus, the increasing density of the dialysate as solute (acid) is transferred to it and the decreasing density of the feed, both serve to facilitate a balanced flow distribution among the cells. Hydrostatic pressure in the dialysate compartment is maintained at a higher level than in the feed cells, which ensures that the membranes are pressed against the support structure of the feed frames (Kessler and Klein, 1992).

Although a great percentage of the dialysis industry is concentrated on hemodialysis, desalination of brackish water, and edible salt from sea water, several other applications have emerged due to environmental constraints placed on industry. Dialysis is an alternative to traditional waste treatments. Some of the recent applications of dialysis for pollution prevention are the recovery of inorganic acids and bases from spent pickling or etching baths. The acid and bases are normally treated by neutralization with the production of sludge and zero recycling of any of
the components. Dialysis offers the possibility to recycle up to 95 percent of the acid or base depending on the type of solution. Typical applications are: recovery of nitric acid from copper nitrate in the jewelry industry, recovery of sulfuric acid from aluminum anodizing, or steel pickling, phosphoric acid from electropolishing and bright dipping applications, hydrofluoric acid from stainless steel and titanium pickling operations, hydrochloric acid from metal cleaning and cationic ion-exchange regeneration solution, and sodium hydroxide from aluminum etching in the aerospace industry (Ondrick, 1993; Graver, 1993). In general we can say that dialysis should be considered when the concentration of the material to be separated is high or when the material to be transferred or the materials to be retained can be changed by harsher separation techniques (Lacey, 1988).

## Dialysis Principles & Definition

Dialysis is defined as the transfer of solute molecules across a membrane by diffusion from a concentrated solution to a dilute solution. A simultaneous diffusion of solvent molecules through the membrane occurs in the opposite direction, a phenomenon called osmosis (Perry and Green, 1984). Dialysis is a diffusion-based separation process that uses a semipermeable membrane to separate species by virtue of their different mobilities in the membrane. Non-electrolyte solute transport across the membrane occurs by diffusion, driven by the difference in solute chemical potential between the two membrane-solution interfaces. The relative transfer of two non-electrolyte solutes across a dialysis membrane is a function of both their diffusivities in the membrane and the driving forces. Separations will be efficient
only for species that differ significantly in the diffusion coefficients. Since diffusion coefficients are a relatively weak function of molecular size, dialysis is limited in practice to separating species that differ significantly in molecular size. As we mentioned before, dialysis is a useful technique only when the solute to be separated is in high concentration. This is because solute fluxes in dialysis are directly dependent on the transmembrane concentration gradient. For this reason, dialysis is considered a slow process, when compared with other membrane separation techniques (Kessler and Klein, 1992). Electrolyte dialysis process can be enhanced by taking advantage of charge repulsion/attraction effects between solute and membrane. Dialysis process in which the membrane has charged ionic groups (ion-exchange membrane) is defined as *Donnan Dialysis*. Donnan Dialysis differs from the classical forms of dialysis in that it relies on ion-selective membranes to retard the transfer of either a cation or anion across the barrier. The charge species that is allowed to cross the membrane will equilibrate across the membrane until the "Donnan equilibrium" conditions are reached (Wallace, 1967; Kessler and Klein, 1992). Writing "aᵢ⁺" and "aᵢ⁺" as the activities of ion "i" in the solutions I and II on opposite sides of a charge membrane, the Donnan equation can be written as:

$$ \left( \frac{a_i^i}{a_i^\infty} \right)^{\frac{1}{z_i}} = K_D \quad (1) $$

where "zi" is the ion valence, and "K_D" is a constant. The Donnan equation applies to any ion that can move through the membrane; as a result, the constant "K_D" serves to equate the indicated ratios of activities of any pair of ions (Cwirko and
Carbonell, 1990). There can be two types of Donnan dialysis depending on the type of membrane used. Donnan dialysis with cation-exchange membrane will allow cations of hydrogen and metals to cross the membrane, but will exclude anions. If for example a solution of nickel sulfate (NiSO₄) is separated from a solution of sulfuric acid (H₂SO₄) by a cation-exchange membrane, the sulfate ions will be excluded from any transport across the membrane, but, the hydrogen and nickel ions will redistribute according to the Donnan equation ratio. Since the concentration of hydrogen ions is much higher in one side of the membrane, the hydrogen will diffuse into the nickel sulfate solution. Since electroneutrality has to be maintained at all times, nickel ions will diffuse into the sulfuric acid solution until Donnan equilibrium is reached, even against a concentration gradient. This type of Donnan dialysis has been named *Donnan Ion-exchange* (Cwirko and Carbonell, 1990). Another case of Donnan dialysis is when the system utilizes an anion-exchange membrane. In this case, anions will be allowed to cross the membrane, but, cations will be excluded with the exception of hydrogen. Hydrogen because of its small size and charge (charge density) is able to cross the anion-exchange membrane, especially with membranes that are made of a weak-base, such as primary and secondary amines (Simon, 1991). This alternative is most often used to separate acids from metal salts.

An example of this type of system is when we have a solution, sulfuric acid mixed with nickel sulfate salt, on one side of the anion-exchange membrane, and pure water or diluted sulfuric acid on the other side. Sulfate ions, which are present in greater numbers than either cation (hydrogen or nickel), will diffuse across the anion-
exchange membrane because of its concentration gradient, and exerts an electrostatic
pull on the cations. Because of its great mobility, hydrogen ions will go along with
the sulfates to maintain electroneutrality, and nickel ions will move across the
membrane at a much lower rate, because of its limited mobility (Lacey, 1988). If
metal cations (nickel) are sufficiently excluded from the membrane, the acid (sulfuric)
can be transported against its concentration gradient. During this process a stream of
pure sulfuric acid can be withdrawn from one compartment (dialysate) and a stream
of purified nickel sulfate can be withdrawn from the opposite compartment (filtrate or
feed). This type of Donnan dialysis has been named *diffusion dialysis*. Donnan
dialysis separations are especially attractive since the separation is driven completely
by **electrochemical potential** gradients across the membrane (Mears, 1983; Cwirko
and Carbonell, 1990).

Transport Phenomena in Dialysis

Mass transfer in dialysis can be divided into two main process: dialysis of non-electrolytes with uncharged membranes, where diffusion will depend on the chemical potential gradient across the membrane, and dialysis of electrolytes with charged membranes, where diffusion will depend on the electrochemical potential gradient across the membrane.

Diffusion of non-electrolytes in uncharged membranes is a spontaneous
process which leads a system to thermodynamic equilibrium by an increase in total
entropy or, when it is conducted isothermally, by a decrease of total free energy.
The gradient of the partial free energy of a diffusing solution component may be
considered to be equivalent of a driving force upon the molecules which causes them
to move against a resistance. The mean velocity of the diffusing molecules may be
considered to be proportional to the acting force, and its given by:

$$v=Uf \quad (2)$$

where "v" is the mean velocity of diffusion, "f" is the acting force, and "U" is a proportionality constant. The force, "f", is associated with the chemical potential gradient, a function which measures the potential energy of a solute molecule. So the true driving force for diffusion is:

$$f = - \frac{d\mu}{dx} = - \frac{RT da}{a dx} \quad (3)$$

(Denbigh, 1981),

where "μ" is the chemical free energy (or chemical potential) per molecule and "a" is the thermodynamic activity. Usually when the solution is considered ideal (highly diluted), the activity is assumed to be equal to the concentration. If we assume an ideal solution then the equations (2) and (3) can be rewritten as:

$$C_v = -RTU \frac{dC}{dx} = -D \frac{dC}{dx} \quad (4)$$

which is an expression of Fick's First Law, where the coefficient "D", is termed the
diffusivity, and "C" is the local non-electrolyte solute concentration within the
membrane. If we express "Cv" in terms of it's equivalent molecular flux, "J_d", in
units of moles per second - square centimeter of traverse area (Tuwiner, 1962) then:
$$J_d = -D \frac{dC}{dx} \qquad (5)$$

Non-electrolyte solute molecules can be transported across the membrane by another phenomena called convective transport or ultrafiltration. If significant ultrafiltration does occur, mass transfer will be enhanced by the addition of a convective component to the diffusive mass transfer. Ultrafiltration will occur from the feed to the dialysate in response to a pressure gradient across the membrane, which may be either applied, or obligatory as a consequence of the geometry of the device and the feed flow rate. Less commonly, if the feed is highly concentrated and the dialysate dilute, ultrafiltration may occur from the dialysate to the feed in response to an osmotic gradient. In the first case, ultrafiltration will enhance solute transport while in the second it will impede it (Klein et al, 1987). Convective transfer can be characterized with the following expression:

$$J_c = J_v(1-\sigma)C \qquad (6)$$

where "$J_c$" is the flux of non-electrolyte solute across the membrane by convection, "$J_v$" is the transmembrane solvent velocity, and "$\sigma$" is the reflection coefficient. The reflection coefficient is a measure of a solute's ability to enter the pores of the membrane. For solutes able to pass the membrane freely, $\sigma = 0$, while for solutes completely excluded by the membrane, $\sigma = 1$.

For homogeneous membranes, the flux "$J$", of a non-electrolyte solute across the membrane at any point is given by the sum of the diffusion and convection transfer:
$$J=J_d+J_c=-D\frac{dC}{dx}+J_v(1-\sigma)C \quad (7)$$

If we integrate equation (7) across the membrane at steady state, we get the overall solute flux "J" at a point on the membrane by the following expression:

$$J=P_m\Delta C+J_v(1-\sigma)C^* \quad (8)$$

Where "P_m" is the membrane permeability, ΔC is the concentration difference across the membrane, and "C^*" is the average concentration in the membrane. There have been several attempts to correlate "P_m" with the membrane structure. Two models are available to determine "P_m" based on membrane properties, the first:

$$P_m=D_s F \frac{V^*}{hl} \quad (9)$$

where "D_s" is the effective solute diffusivity coefficient in the solution, "F" is the drag factor, "V^*" is the fraction of membrane volume occupied by pores, "h" is the tortuosity, and "l" is the wet membrane thickness (Perry and Green, 1984). Another alternative to estimate the permeability is based on the hypothetical pore structure, and an experimental method to determine the ratio between the fractional cross sectional area "Ap" and the actual pore path length "lp", the relationship between permeability "P_m" and the "Ap/lp" is expressed by:

$$P_m = \frac{\left[ \frac{\Delta p}{lp} \right] D_s [1-q]^2}{K_1} \quad (10)$$

Where "q" is the ratio of solute to pore diameter, and "K_1" is a power series in "q".
Yasuda and coworkers (1968, 1969, 1971) found the following relationships between permeability and the membrane structure: 1) The permeability of solutes, whose size is small compared to the membrane pore size, is proportional to the degree of membrane hydration. 2) Membrane permeability decreases exponentially with increasing molecular size, where the later is expressed in terms of molecular cross-sectional area. 3) solute reflection coefficients change markedly when solute size approaches the average pore size of the membrane (Klein et al, 1987).

As solute is removed from the feed-side membrane-solution interface by dialysis, the layer is depleted and its concentration must be restored from the bulk solution. In laminar flow, which is usual in dialysis, there is no convection and regeneration of the interfacial layer is solely by diffusion from the bulk solution. As such diffusion occurs, the concentration gradient from the bulk solution to the interfacial layer decreases. The same situation is present on the dialysate side of the membrane. As solute crosses the membrane, it must be transported away from the dialysate-side interfacial layer to maintain the transmembrane concentration difference. Both boundary layers are a function of feed or dialysate flow rate, transmembrane flux, and solute diffusivity in the solvent. These solute concentration boundary layers may contribute significantly to the overall resistance, if the membrane is thinner than the boundary layers (Klein et al, 1987; McCabe et al, 1993). The contribution of the boundary layer to overall mass transfer resistance suggests the dialyzer should be operated with flows in the turbulent flow region to minimize boundary-layer formation. In some cases, the dialysate side of the membrane can be operated in the
turbulent flow region, especially in hemodialysis applications with hollow-fiber units, but, in other applications like waste-water treatment, the dialysate flows in the laminar flow region. In any application the feed is always operated with flows in the laminar flow region. For laminar flow in the absence of ultrafiltration, the pressure drop, "ΔP", on the feed side is given by the Hagen-Poiseuille equation:

$$ \Delta P = \frac{128 \mu Q_F l}{\pi N d^4} \quad (\text{for hollow-fibers}) \quad (11) $$

$$ \Delta P = \frac{12 \mu Q_F l}{Nwh^3} \quad (\text{plates}) \quad (12) $$

where "μ" is the viscosity of a fluid flowing at a rate "Q<sub>F</sub>" through a dialyzer containing "N" channels, or tubes, of diameter "d", or depth "h", with "w", and length "l". If we calculate the pressure drop under turbulent flow conditions, the pressure drop obtained mandates a feed pressure which leads to convection into the dialysate (ultrafiltration), particularly for highly permeable membranes. In addition, an undue mechanical stress may be placed on the membrane. In hemodialysis, the flow must be kept in the laminar flow region to avoid undue mechanical stress on the blood cells which may result in deterioration or destruction (Klein et al, 1987). An overall mass transfer coefficient, "K", is calculated from the sum of the mass transfer resistance in the membrane and the two boundary layers:

$$ \frac{1}{K} = \frac{1}{P_m} + \frac{1}{k_f} + \frac{1}{k_d} \quad (13) $$

where "k<sub>f</sub>" and "k<sub>d</sub>" are the mass transfer coefficient for the concentration
polarization layers (boundary layers) on the feed and dialysate sides of the membrane, respectively. Membrane permeability "P<sub>m</sub>", which is proportional to solute diffusivity "D" in the membrane, will become the dominant factor contributing to "K" as solute molecular size increases. However, for small solutes, the membrane permeability will limit mass transfer only when "K<sub>f</sub>" and "K<sub>d</sub>" are very large relative to "P<sub>m</sub>". For approximations "K" can be estimated theoretically, but for design purposes it is almost always necessary to determine "K" experimentally (Lacey, 1988; Klein et al, 1987).

In the dialytic process, the transmembrane solvent velocity "J<sub>v</sub>" is generally less than 3 x 10<sup>-5</sup> cm/s and permeability coefficients are of the order of 10<sup>-4</sup> cm/s.

When high transmembrane pressures are obligatory because of flow constraints, the value of "J<sub>v</sub>" may increase, and convection may become an important contribution to overall mass transfer (Klein et al, 1987). Assuming a negligible convective contribution, the mass transfer of a given non-electrolyte solute "dm" across a differential element of area "dA" can be estimated using the overall mass-transfer

coefficient, equation (7) can be modified to give:

$$J = \frac{dm}{dA} = K \frac{dC}{dx} \qquad (14)$$

If the concentration difference between the two liquids "dC/dx" is:

Equation (8) becomes:
$$ \Delta C = C_F - C_D \qquad (15) $$

$$ J = \frac{dm}{dA} = K [C_F - C_D] \qquad (16) $$

where "C<sub>F</sub>" and "C<sub>D</sub>" are the solute concentration in the feed and dialysate at an specific point of the membrane, respectively. If we integrate equation (16) along the length of the membrane the following equation for the overall mass transfer is obtained:

$$ m = KA\Delta C_{in} \qquad (17) $$

where

$$ \Delta C_{in} = \frac{(C_{Fo} - C_{Di}) - (C_{Fi} - C_{Do})}{\ln[(C_{Fo} - C_{Di})/(C_{Fi} - C_{Do})]} \qquad (18) $$

and "C<sub>Fo</sub>", "C<sub>Fi</sub>", "C<sub>Do</sub>", and "C<sub>Di</sub>" are the feed and dialysate concentrations at the outlet and inlet, respectively (Mears, 1983; Lacey, 1988; Klein et al, 1987).

If convective transport is not negligible, equation (8) has to be integrated over the total length of the membrane. Expansion of equation (8) from a point on the membrane to the dialyzer as a whole requires that the feed and dialysate solute concentrations at a point on the membrane be expressed in terms of known concentrations. Schindhelm, Farrell, and Stewart (1977) developed an expression based on zero solute concentration at the dialysate inlet, and a linear dialysate concentration profile in a countercurrent dialyzer for solutes greater than 300 dalton molecular weight (Klein et al, 1987).
$$D^* = \frac{X}{1 - Y} \qquad (19)$$

where

$$X = - \frac{KA_m + Q_{uf} (1-\sigma)}{Q_{fi} - Q_{uf}} \frac{Q_{fi}}{Z} [1 - \exp(Z)] \qquad (20)$$

$$Y = -[KA_m + Q_{uf}(1-\sigma)] \frac{KA_m}{Q_{fi}Q_{do}Z^2} [1 + (\frac{Z}{2}) + (\frac{1}{Z})(1-\exp(Z))] + \frac{X}{Q_{fi}} + \frac{KA_m}{ZQ_{do}} \qquad (21)$$

$$Z = \frac{KA_m - Q_{uf} \sigma}{Q_{fi}} \qquad (22)$$

where "$A_m$" is the membrane area, "$Q_{uf}$" is the ultrafiltration rate from the feed to the dialysate, "$Q_{fi}$" is the feed flow rate at the inlet, "$Q_{do}$" is the dialysate flow rate at the outlet, and "$D^*$" is the dialysance defined as the rate of mass transfer divided by the concentration difference between inlet feed and inlet dialysate. Under this model as ultrafiltration increases (feed into dialysate direction) from 0 to 0.5 cm³/s, a 200 dalton molecular weight solute increases its transfer by 12%, while a 2000 dalton molecular weight solute increases its transfer by 84%. This demonstrates the importance of convective transport for large-molecular-weight solutes (Kessler and Klein, 1982). The difference in the effect of convective transport over large or small solutes depends upon the permeability and reflection coefficient of small and large molecules. As the molecular weight of a solute increases the permeability decreases logarithmically while the reflection coefficient decreases linearly (Wendt et al, 1979).

Diffusion of electrolytes in uncharged membranes, is driven by the
electrochemical potential between the two membrane-solution interfaces. As we mentioned before, when two phases are in equilibrium at the same temperature and pressure, the chemical potential of every component which can be transferred from one phase to the other will be the same in each phase. This definition of chemical equilibrium ignores electrical forces, or assumes the component transferred is electrically neutral (non-electrolytes). In order to find the electrochemical potential difference between two phases, it is necessary to consider infinitesimal transfer of ions from one phase to the other. If "μ₁" and "μ₂" are the chemical potentials of an ion in two phases which are in equilibrium with each other, "V₁" and "V₂" are the electrical potentials in the two phases, "z" is the valency of the ion. The electrical charge per gram of ion is "zF" and "dn" are the moles of ion transferred from phase 1 to phase 2, the change in free energy of the system is:

$$ \Delta G = -\mu_1 \, dn + \mu_2 \, dn \qquad (23) $$

The electrical work "We" in transferring a charge "zF dn" from a region of potential "V₁" to a region of potential "V₂" is:

$$ We = (V_2 - V_3) zF \, dn \qquad (24) $$

The total work done in the transfer is thus:

$$ W_t = (\mu_1 \, dn + \mu_2 \, dn) + (V_2 - V_1) \, zF \, dn \qquad (25) $$

Applying the conditions for equilibrium, equation (25) becomes zero, and if "dn" is
different than zero, we have:

$$ \mu_2 - \mu_1 + (V_2 - V_1) zF = 0 \qquad (26) $$

$$ \mu_2 + V_2 zF = \mu_1 + V_1 zF \qquad (27) $$

Therefore, for equilibrium of ions in different phases, the quantity $(\mu + zFV)$ for each ion must be the same in each phase. This quantity is called "electrochemical potential" (Lakshminarayanaiah, 1969).

The rate of diffusion or flux "J" of ions across a porous plug (uncharged membrane) is given by the product of three terms: a proportionality constant, concentration "$C_i$", and force. The flux due to diffusion "$J_d$", caused by the force represented by the gradient of chemical potential, is given by:

$$ J_d = \frac{D_i}{RT} C_i \left(-\frac{d\mu_i}{dx}\right) = -D_i C_i \frac{d \ln a_i}{dx} \qquad (28) $$

where "$D_i$" is the diffusivity of ion "i". If we substitute the activity "$a_i$" for ($a_i = C_i \gamma_i$) equation (28) becomes:

$$ J_d = -D_i C_i \left[ \frac{d \ln C_i}{dx} + \frac{d \ln \gamma_i}{dx} \right] = -D_i \left[ \frac{d C_i}{dx} + C_i \frac{d \ln \gamma_i}{dx} \right] \qquad (29) $$

When "$\gamma_i$" is constant (diluted solutions), the flux equation reduces to Fick's first law.

$$ J_d = -D_i \frac{dC_i}{dx} \qquad (5) = (30) $$

The diffusion of charged species like ions, generates an electric field which acts as another driving force. Thus the flux due to the generated electric field "$-dE/dx$" is
given by:

$$J_E = -\frac{u_i z_i C_i dE}{dx} \qquad (31)$$

where "u_i" is the ion mobility, which is related to the diffusivity "D_i" by

$$D_i = \left(\frac{RT}{F}\right)u_i \qquad (32)$$

The total flux "J" is given by:

$$J = J_d + J_E = -\frac{RT}{F} u_i \left[ \frac{dC_i}{dx} + C_i \frac{d \ln \gamma_i}{dx} + z_i \frac{F}{RT} C_i \frac{dE}{dx} \right] \qquad (33)$$

This is the Nernst-Planck flux equation. It applies to all mobile species and the set of flux equations, one for each species, is subject to the condition of electroneutrality:

$$\sum z_i C_i = 0 \qquad (34)$$

At steady state, when the concentrations are independent of time, the equation of continuity (Fick's second law):

$$\frac{\partial C}{\partial t} = \frac{\partial}{\partial x} (D \frac{\partial C}{\partial x}) \qquad (35)$$

becomes:

$$\frac{\partial C_i}{\partial t} = -\nabla J_i = 0 \qquad (36)$$
The total current "I" carried by the charge species is given by:

$$I = F \sum z_i J_i \quad (37)$$

Assuming ideal solution (γ = 1) equation (33) becomes:

$$J_i = -\frac{RT}{F} u_i \left( \frac{dC_i}{dx} + z_i \frac{F}{RT} C_i \frac{dE}{dx} \right) \quad (38)$$

substituting equation (38) into equation (37), we obtain the current "I" carried by a flux of ion "i"

$$I_i = -z_i \mu_i \left( RT \frac{dC_i}{dx} + z_i F C_i \frac{dE}{dx} \right) \quad (39)$$

Equation (39) together with the equation of continuity (36) gives for all positive and negative ions (univalent)

$$\frac{\partial C_i^+}{\partial t} = u_i^+ \left[ RT \frac{\partial^2 C_i^+}{\partial x^2} + F \frac{\partial}{\partial x} \left(C_i^+ \frac{\partial E}{\partial x}\right) \right] \quad (40)$$

$$\frac{\partial C_i^-}{\partial t} = u_i^- \left[ RT \frac{\partial^2 C_i^-}{\partial x^2} - F \frac{\partial}{\partial x} \left(C_i^- \frac{\partial E}{\partial x}\right) \right] \quad (41)$$

Since

$$\sum C_i^+ = \sum C_i^- = C \quad (42)$$

and
$$ \sum \frac{\partial C_i^+}{\partial t} - \sum \frac{\partial c_i^-}{\partial t} = 0 \qquad (43) $$

Subtracting equation (40) and (41) we obtain:

$$ \sum u_i^- RT \frac{\partial^2 C_i^+}{\partial x^2} - \sum u_i^- RT \frac{\partial^2 C_i^-}{\partial x^2} + \sum u_i^+ F \frac{\partial}{\partial x} (C_i^+ \frac{\partial E}{\partial x}) + \sum u_i^- F \frac{\partial}{\partial x} (C_i^- \frac{\partial E}{\partial x}) = 0 \quad (44) $$

integrating equation (44) gives:

$$ \sum u_i^+ RT \frac{\partial C_i^+}{\partial x} - \sum u_i^- RT \frac{\partial C_i^-}{\partial x} + \sum u_i^+ FC_i^- \frac{\partial E}{\partial x} + \sum u_i^- FC_i^+ \frac{\partial E}{\partial x} = 0 \quad (45) $$

The integration constant is zero, as the partial derivatives vanish when "Ci" is constant in the interval "Δx".

Substituting "$\sum u_i + C_i =$ U" and "$\sum u_i - C_i =$ V" equation 45 becomes:

$$ RT \frac{\partial U}{\partial x} - RT \frac{\partial V}{\partial x} + FU \frac{\partial E}{\partial x} + FV \frac{\partial E}{\partial x} = 0 \qquad (46) $$

Integration of equation (46) leads to the following expression:

$$ \frac{\xi U_2 - U_1}{V_2 - \xi V_1} = \frac{\ln(C_2/C_1) - \ln\xi}{\ln(C_2/C_1) + \ln\xi} \frac{\xi C_2 - C_1}{C_2 - \xi C_1} \quad (47) $$

where the total diffusional electrical potential "$E_L$" is equal:

$$ E_L = \frac{RT}{F} \ln \frac{\gamma_1}{\gamma_2} = \frac{RT}{F} \ln \xi \qquad (48) $$

where "$\gamma_1$ and $\gamma_2$" are the equivalent conductances of the two solutions at the same concentration "C". Planck's integration is applicable to solutions containing univalent ions or ions of the same numerical valency (Lakshminarayanaiah, 1969). Jonsson
(1904) has given an integrated solution for a mixture of salts in which all cations have one valency "Z_c" and all anions valency "z_a":

$$ \frac{U_2\xi^{z_c} - U_1}{V_2 - \xi^{z_c}V_1} = \frac{z_a \ln(C_2/C_1) - z_a \ln\xi}{z_c \ln(C_2/C_1) + z_c \ln\xi} \frac{C_2\xi^{z_c} - C_1}{C_2 - \xi^{z_c}C_1} \quad (49) $$

Any barrier which is called a membrane may be "neutral" but in reality it is difficult to find one without carrying some negative or positive fixed charges. The neutral membrane is used in forming the so-called "constrained diffusion" boundary used for the previous derivation postulated by Planck (Lakshminarayanaiah, 1969).

Diffusion of electrolytes in charged membranes is driven by the electrochemical potential between the two membrane-solution interfaces, plus the attraction or repulsion forces between the membrane and the electrolyte. When an electric field acts on an electrolyte solution, cations move to the cathode and anions move to the anode. The moving ion transfers its momentum to the surrounding medium which is mostly water. When a crosslinked three dimensional matrix membrane is interposed, dividing the solution chamber into two compartments and acting selectively to certain ions, the existence of charged groups allows the membrane to have more counterions than co-ions. As a result, the appearance of an electric field gradient across the membrane allows the counterions to impart more momentum to solvent molecules than do co-ions, resulting in a unidirectional flow of solvent. In the case of a positively-charged membrane, anions which are the counterions, migrate to the anode toward which solvent will also flow. The counterions will experience less resistance to flow than do co-ions. Consequently the
migration or convection of the pore liquid will increase the flux of the counterions.

This flux may be written as:

$$J_c = v \cdot C \qquad (50)$$

where "v" is the velocity of movement of the center of gravity of the pore liquid, and "C" is the local electrolyte solute concentration within the membrane. As we can see, this equation is very similar to equation (6) with the exception of the rejection coefficient. By adding the convection and the pressure difference flux to equation (33) we obtain the extended form of the Nernst-Planck flux equation given by Schlögl (1966)

$$J = C_i v^* - D_i \left[ \frac{dC_i}{dx} + C_i \frac{d \ln \gamma_i}{dx} + z_i \frac{F}{RT} C_i \frac{dE}{dx} + \frac{C_i}{RT} \left( V_i - \frac{M_i}{M_w} V_w \right) \frac{dP}{dx} \right] \quad (51)$$

where "i" and "w" refer as dissolved species and solvent respectively, "M_i" is the species molecular weight, "V_i" is the species partial molar volume, "V_w" is the solvent partial molar volume, and "P" is the hydrostatic pressure in the pore medium. This equation is applicable to ideal cases where there are no molecular or ionic interactions, it applies to all mobile species and is governed by the conditions of electroneutrality and continuity (Lakshminarayanaiah, 1969). Integration of equation (51) is given by Schlögl (1966), but recently Cwirko and Carbonell (1990) have reduced equation (51) to a dimensionless ordinary differential equation using the techniques of area averaging. Equation (52) is the area-averaged axial component of the extended Nernst-Planck equation, valid for dilute solutions of electrolytes with constant ion diffusivities, the area-averaged continuity equation for the electrolyte "i"
$$ \langle \bar{J}_i \rangle = \langle \bar{C}_i \rangle Pe + \langle \overline{C_i v_x} \rangle - \frac{D_i}{D_1} \left[ \frac{d\langle \bar{C}_i \rangle}{dx} + z_i \langle \bar{C}_i \rangle \frac{d\langle \bar{E} \rangle}{dx} + \langle \overline{C \cdot \partial E / \partial x} \rangle \right] \quad (52) $$

and the solvent respectively

$$ \frac{d\langle \bar{J}_i \rangle}{dx} = 0 \qquad (53) $$

$$ \frac{dPe}{dx} = 0 \qquad (54) $$

from the Navier-Stokes equation, the Peclet number can be obtained by:

$$ Pe = - \frac{d \langle \bar{P} \rangle - \langle \bar{\pi} \rangle}{dx} - \beta \sum \left[ \left( \frac{d \langle \bar{c}_i \rangle}{dx} + z_i \langle \bar{c}_i \rangle \frac{d \langle \bar{E} \rangle}{dx} \right) (1 + \langle A_i \rangle) + \langle B_i \rangle \right] \quad (53) $$

the overall electroneutrality in the membrane pore can be expressed as

$$ \frac{C_m}{C_i} + \sum \langle \bar{C}_i \rangle = 0 \qquad (56) $$

and the area-averaged current through a pore is given by

$$ \langle \bar{i} \rangle = \sum z_i \langle \bar{J}_i \rangle = 0 \qquad (57) $$

where "Ci" is the dimensionless electrolyte concentration in the membrane pore, "Cm" is the number of fixed pore surface charges per pore volume, "vx" is the dimensionless area-averaged axial velocity of the solvent, "E" is the dimensionless electrical potential, "π" is the dimensionless osmotic pressure, "β" is the dimensionless group affecting the osmotic flow through the pore, "Aᵢ" and "Bᵢ" are dispersion terms. In order to predict ion fluxes through a charged porous membrane
using this capillary tube model, it is necessary to know the average pore radius, the average pore length, the average pore surface, and the pore area per unit of total membrane area (Cwirko and Carbonell, 1990).

## Dialysis as a Unit Operation

Dialysis can be used as a unit operation in two basic configurations, as a batch process or in a continuous process stream. Batch dialysis is widely use in hemodialysis applications, where a feed reservoir of volume "V" (body water) with a solute concentration "C<sub>f</sub>", is dialyzed continuously with an inlet concentration equal to "C<sub>di</sub>". The feed and dialysate streams enter the dialyzer with flows of "Q<sub>f</sub>" and "Q<sub>d</sub>", respectively. The main difference between continuous and batch dialysis is, that, in a batch system the feed outlet stream returns to the reservoir while in continuous systems does not. In the case of the dialysate, the stream runs continuously without recycling it back into the dialyzer. A mass balance for a batch system can be written as follows:

$$ \frac{d (VC_f)}{dt} = Q_d (C_{do} - C_{di}) \quad (58) $$

if "D*" is the mass transfer rate divided by the concentration difference between inlet feed and inlet dialysate as:

$$ D^* = \frac{Q_d (C_{do} - C_{di})}{C_f - C_{di}} \quad (59) $$

equation (58) becomes

substituting "V" by $V_o - Q_{af} \cdot t$, where "Q<sub>af</sub>" is the rate of ultrafiltration from the feed
$$ \frac{d (VC_f)}{dt} = -D \cdot (C_f - C_{di}) \quad (60) $$

to the dialysate and "Vo" is the initial feed volume, equation (60) becomes

$$ (V_o - Q_{uf} \cdot t) \frac{dC_f}{dt} - Q_{uf}C_f = -D \cdot (C_f - C_{di}) \quad (61) $$

integrating equation (61) gives us the equation to obtain the reservoir concentration with respect to time

$$ C_f(t) = \frac{D \cdot C_{di}}{D^8 - Q_{uf}} \left[ 1 - \left( \frac{V_o - Q_{uf} \cdot t}{V_o} \right)^b \right] + C_f^o \left( \frac{V_o - Q_{uf} \cdot t}{V_o} \right)^b \quad (62) $$

where "C<sub>of</sub>" is the initial feed solution concentration and

$$ b = \frac{(D \cdot Q_{uf})}{Q_{uf}} \quad (63) $$

(Kessler and Klein, 1992).

In a single-time-pass continuous dialysis system, the change in concentration with respect to time in both streams, feed and dialysate, is equal to zero assuming steady state conditions. Batch or continuous dialysis performance is measured based on the percentage of solute recovered in the dialysate, percentage of solute concentration in the dialysate and percentage of rejected solute leakage in the dialysate. The percentage of recovered solute is obtained by the ratio between the amount of solute in the dialysate divided by the amount of solute in the feed
$$ \% Rec = \frac{V_{do}C_{do}}{V_{f_j}C_{f_j}} = \frac{V_{do}C_{do}}{V_{f_j}C_{f_j} + V_{do}C_{do}} \quad (64) $$

where "$V_{do}$" is the volume in the dialysate, "$V_{f_j}$" is the volume of feed introduced in the dialyzer, "$V_{f_j}$" is the volume of depleted feed leaving the dialyzer, "$C_{do}$" is the solute concentration in the dialysate, "$C_{f_j}$" is the solute concentration in the feed, and "$C_{f_j}$" is the solute concentration in the depleted feed. This ratio indicates how much solute is been separated compared to the total amount.

The percentage of solute concentration can be obtained by the ratio of dialysate concentration to feed concentration:

$$ \%C = \frac{C_{do}}{C_{f_j}} \quad (65) $$

This ratio gives us information on the dilution magnitude of our dialysate compared to the initial feed. As we mentioned before, there are several ways on which the dialysate can be diluted or concentrated according to the water flow across the membrane and water flow rate in the dialysate.

Finally, the percentage of rejected solute " % Leakage " can be calculated as the ratio between the amount of rejected solute in the dialysate divided by the amount of rejected solute in the feed

$$ \%L = \frac{V_{do}C_{do}}{V_{f_j}C_{f_j}} = \frac{V_{do}C_{do}}{V_{do}C_{do} + V_{f_j}C_{f_j}} \quad (66) $$

where "$C_{do}$", "$C_{f_j}$", and "$C_{f_j}$" are the concentrations of the rejected solute in the
dialysate, inlet feed, and depleted feed respectively. This ratio indicates how much unwanted solute is crossing the membrane (leaking) compared to the total amount in the feed.

Several factors affect the performance of dialysis separation, the most important ones are: concentration difference across the membrane, temperature, solute selectivity, pressure difference across the membrane, and streams flow rates. The greater the difference in concentration across the membrane, the greater the rate of diffusion is. The concentration difference can be modified based on the initial feed and dialysate inlet concentrations as well as flow rates. High dialysate and low feed flow rates increase the concentration difference across the membrane (Sridhar and Subramaniam, 1989). When using a plate-frame dialyzer, countercurrent flow produces higher concentration difference along the dialyzer than cocurrent. The higher hydrostatic pressure difference across the membrane, the higher the transport of solute molecules with the solvent flow. Higher hydrostatic pressure in the feed increases the flow of water into the dialysate (reverse osmosis), this flow enhanced the transfer of solute into the dialysate. The higher the selectivity of the membrane, the higher the diffusion electrical potential difference across the membrane, which increases the flux of solute. These three factors can be identified as major components of the extended Nernst-Planck flux equation, equation (51). The last factor is the temperature, higher temperature increases the solute flux by increasing the solute diffusivity (Bailey, 1989; Sato et al, 1985).
REFERENCES

Bailey, D.E., Recovery and Reuse of Spent Acid Baths via Diffusion Dialysis, *Hazardous and Municipal Waste Minimization Conference, Providence RI Feb.*
7-8, 1989.

Cwirko, E.H. and Carbonell, R.G., A Theoretical Analysis of Donnan Dialysis Across Charged Porous Membranes, *Journal of Membrane Science*, 1990, 48, 155-179.

Davis, T.A., Application of Ion-Exchange Membranes to Industrial Separations, *Consultant Information*, NJ 1992, 1-7.

Denbigh, K., *The Principles of Chemical Equilibrium*, Fourth ed., Cambridge University Press, London 1982, 116-119

Drukker, W., Parsons, F.M., and Maher, J.F., *Replacement of Renal Function by Dialysis*, 2nd ed., Martinus Nijhoff, Boston MA 1983.

Graver Water, Division of The Graver Company, *Membrane Technologies Difusion Dialysis*, Union NJ 1993.

Jonsson, G., *Synthetic Membranes: Science, Engineering and Applications*, ed. P.M. Bungay, H.K. Lonsdale and M.N. de Pinho, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo 1986, 625-645.

Kessler, S.B. and Klein, E., *Membrane Handbook*, ed. W.S. Winston Ho and K.K. Sirkar, Van Nostrand Reinhold, NY 1992, 161-215.

Klein, E., Ward, R.A. and Lacey, R.E., *Handbook of Separation Process Technology*, ed. R.W. Rousseau, John Wiley & Sons, NY 1987, 954-981.

Kolff, W.J., and Berk, H.T., *The Artificial Kidney: A Dialyzer with a Great Area*, Acta Medica Scandinavica, 1944, 117, 121-134.

Lacey, R.E., *Handbook of Separation Techniques for Chemical Engineers*, ed. P.A. Schweitzer, McGraw-Hill, NY 1988, 1_479-1_495.

Lakshminarayanaiah, N., *Transport Phenomena in Membranes*, Academic Press, NY & London 1969, 58-361.

McCabe, W.L., Smith, J.C. and Harriton, P., *Unit Operations of Chemical Engineering*, McGraw-Hill, NY 1993, 860-863.
Mears, P., Synthetic Membranes: Science, Engineering and Applications, ed. P.M. Bungay, H.K. Lonsdale and M.N. de Pinho, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo 1986, 169-179.

Ondrick, E., Use of Diffusion Dialysis to remove Metals from Nickel Plating Solutions at a Nickel Plating Job Shop, *New England Water Environment Association Zero Discharge Conference*, New Hampshire May 27, 1993.

Perry, R.H., and Green, D.W., *Perry's Chemical Engineers' Handbook*, 6th ed, McGraw-Hill, NY 1984, 17_18-17_26.

Sato, J., Onuma, M., Motomura, H. and Noma, Y., Recovery of Nitric Acid and Hydrofluoric Acid from the Pickling Solution by Diffusion Dialysis, Tokuyama Soda Co., LTD, Reprint from Jitsumu Hyomen Gijutsu, *The metal Finishing Society of Japan* 1985, 1-8.

Scott, J., *Membrane and Ultrafiltration Technology*, Recent Advances, Noyes Data Corporation, NJ 1980, 210-211.

Schindhelm, K., Farrel, P.C., and Stewart, J.H., Convective Mass Transfer in Artificial Kidney, Paper read at 2nd Australian Conference on Heat and Mass Transfer, 16-18 February, 1977, University of Sydney.

Schlögl, R., *Berichte der Bunsengesellschaft fuer Physikalische Chemie*, 1966, 71, 755.

Simon, G.P., *Ion Exchange Training Manual*, Van Nostrand Reinhold, NY 1991, 40.

Sridhar, P., and Subramaniam, G., Recovery of Acid From Cation Exchange Resin Regeneration Waste by Diffusion Dialysis, *Journal of Membrane Science*, 1989, 45, 273-280.

Tuwiner, S.B., *Diffusion and Membrane Technology*, Reinhold Publishing Corporation, NY & London 1962, 38-60, 77-90, 137-151, 287-320.

Wallace, R.M., Concentration and Separation of Ions by Donnan Membrane Equilibrium, *Industrial and Engineering Chemistry, Process Design and Development*, 1967, 6, 423-431.

Wendt, R.P., Klein, E., Bressler, H.E., Holand, F.F., Serino, R.M., and Villa, H., Sieving Properties of Hemodialysis Membranes, *Journal of Membrane Science*, 1979, 5, 23-49.
Yasuda, H., Peterlin, A., Colton, C.K., Smith, K.A., and Merril, E.W.
Permeability of Solutes Through Hydrated Polymer Membranes, III.
theoretical background for selectivity of dialysis membranes, *Die
Makromolkulare Chemie*, 1969, 126, 177-186.

Yasuda, H., Lamaze, C.E., and Ikenberry, L.D., Permeability of solutes
Through Hydrated Polymer Membranes., I. diffusion of sodium chloride,
*Die mMakromolekulare Chemie*, 1968, 118, 19-35.

Yasuda, H., Lamaze, C.E., and Peterlin, A., Diffusive and Hydraulic
Permeabilities of Water in Water-swollen Polymer Membranes, *Journal of
Polymer Science*, 1971, 9, 1117-1131.
<figure><img src="image_2.png" /><figcaption>Figure 1</figcaption></figure>

Countercurrent Diffusion Dialyzer
# Chapter II
# MATERIALS AND METHODS

## Preparation of required solutions

All solutions were made with analytical reagents from Fisher Scientific and deionized (DI) water. DI water was obtain using a milli-Q water system (Millipore), fed from a still (Barnstead). Nitric acid solutions were made with CAS Reg. 7697-37-2 nitric acid (69-71%) and DI water. Sulfuric acid solutions were made with CAS Reg. 7664-93-9 sulfuric acid (98.08%) and DI water. Sodium hydroxide solutions were made with CAS Reg. 1310-73-2 sodium hydroxide (beads) and DI water.

Copper nitrate solutions were made with CAS Reg. 19004-19-4 copper (II) nitrate hemipentahydrate crystalline and DI water. Ferrous sulfate solution were made with CAS Reg. 7782-63-0 iron (II) sulfate heptahydrate crystalline and DI water. Nitric acid / copper nitrate solutions were made with nitric acid solution and CAS Reg. 7440-50-8 copper powder. Sulfuric acid / ferrous sulfate solutions were made with sulfuric acid solution and CAS Reg. 7439-89-6 iron electrolytic powder.

## Diffusion dialysis equipment

A mono-membrane plate and frame diffusion dialysis system was built using specifications for a DD unit model TSD-5, from Tokuyama Soda Co., LTD (1991).

A plexi-glass stack was made to fit a TSD-5 Neosepta AFN membrane, the stack was covered with a polyethylene sheet to prevent the working acid from attacking the plexi-glass. Two gaskets made out of neoprene were used to separate the membrane from the walls, the gaskets held a polyethylene frame. Membranes, gaskets, and
frames were obtained from Tokuyama Soda Co., LTD. Membrane specifications are as follows:

Grade AFN

Type Strongly basic, anion permeable, chloride form

Properties Diffusion dialysis, resistant against organic fouling

Electric Resistance 0.4 - 1.5, equilibrated with 0.5 N NaCl, at 25 C

## Transport Number

Total Anion > 0.98

Na⁺ + K⁺ < 0.02

Ca⁺² + Mg⁺² < 0.02

Cl⁻ > 0.98

SO₄⁻² > 0.98

Burst Strength 2 - 5 (Kg/cm²)

Water Content 0.40 - 0.55 (g.H₂O / g.Na-form dry membrane), equilibrated with 0.5 N NaCl

Exchange Capacity 2.0 - 3.5 (meq/g.Na-form dry membrane)

Thickness 0.15 - 0.20 (mm)

Size 21 x 50 cm

Two diaphragm pumps from LMI Liquid Metronic Division model A341-152S, were used to operate the acid and water streams. Teflon PTFE pipe (3/8") from
Cole-Parmer was used for piping the fluids. A 5 gal polypropylene Nalgene tank (8"x8"x8") was used as a water reservoir, a 500 ml graduated glass cylinder with an outlet at the bottom was used as an acid / metal (feed) reservoir, 125 ml glass bottles were used to collect recovered acid and depleted feed (waste). A plexi-glass tank was built as a temperature control bath to maintain the stack at constant temperature, a Thermomix heater (model 1419 from Braun) was used to maintain the bath temperature constant. A PVC heat exchanger was built to provide constant temperature water, before entering the membrane stack (dialyzer). A Polyscience heater (model 70) provided the exchanger with an appropriate temperature. A magnetic stirrer (Fisher Scientific model 220 T) was used to homogenize the feed when necessary.

## Diffusion dialysis setup

The experiments can be divided into 4 different groups, depending on the variable tested. Every experiment was performed at least two times, or until consistent data was obtained.

1.) The variation in acid concentration was performed at constant: flow rates (feed and water at 50 ml/hr), temperature (water 40 °C and feed at 30 °C), recovery outlet level (41.5 cm), and metal concentration. The setup in Figure 1, was designed to work continuously with an increasing acid concentration in the feed stream. The feed concentration at any time during the experiment can be determined by the following equation (67)
$$N_f = N_{\text{stock}} - N_{\text{stock}} e^{-t/Q_f / V_o J_t} \quad (67)$$

where "$N_f$" is the feed normality, "$N_{\text{stock}}$" is the normality of the stock solution, "$Q_f$" is the feed flow rate, "$t$" is the operating time, and "$V_o$" is the cylinder volume. Experiments were run for approximately 12 h, from which, the first 4 h were not sampled because of low acid concentration in the feed, the remaining 8 h were collected at intervals of 1 hr.

2.) The variation in metal concentration was performed at constant: flow rates (feed and water at 50 ml/hr), temperature (water 40 °C and Feed 30 °C), recovery outlet level (41.5 cm), and acid concentration. The setup in Figure 1, was again used to work continuously with an increasing metal concentration in the feed stream. The metal concentration at any time during the experiment can be determined by the following equation (68)

$$F_{\text{ppm}} = \text{STOCK}_{\text{ppm}} - \text{STOCK}_{\text{ppm}} e^{-t/Q_f / V_o J_t} \quad (68)$$

where "$F_{\text{ppm}}$" is the metal concentration in the feed in ppm units, and "$\text{STOCK}_{\text{ppm}}$" is the metal concentration in the stock solution in ppm. Experiments run for approximately 12 h, the first 4 h were neglected because of low metal concentration in the feed, the remaining 8 h were collected at intervals of 1 hr.

3.) The variation in recovery outlet level was performed at constant: flow rates, temperature, acid concentration, and metal concentration. The setup in Figure 2, was designed to work continuously with the ability to modify the length of the recovery outlet, a better view of recovery outlet level is shown in Figure 3. The maximum
level is 41.5 cm, which corresponds to the feed inlet. Lowering the recovery outlet
level decreases the column of fluid which reduces the hydrostatic pressure. By
reducing the recovery hydrostatic pressure, we were able to reduce the pressure
difference across the membrane, "ΔP = P<sub>rec</sub> - P<sub>f</sub>". The feed pressure "P<sub>f</sub>", was
calculated according to the formula "P<sub>f</sub> = ρ<sub>w</sub> g h<sub>w</sub>", where "ρ<sub>w</sub>" is the waste solution
density, "g" is the gravitational acceleration, and "h<sub>w</sub>" is the feed side column length
(41.5 cm). The recovery hydrostatic pressure "P<sub>rec</sub>", was calculated according to the
formula "P<sub>rec</sub> = ρ<sub>rec</sub> g h<sub>rec</sub>", where "ρ<sub>rec</sub>" is the recovered acid density, and "h<sub>rec</sub>" is
the recovery outlet length. Experiments ran for approximately 12 h, the first 4 h
were needed to reach steady state, the remaining 8 h were sampled at intervals of 1
hr. The recovery outlet level was adjusted every hour after the steady state period,
always starting from 41.5 cm.

4.) The variation in feed temperature and water temperature were performed at constant: acid concentration, metal concentration, flow rates, recovery outlet level, and water or feed temperature, depending on which temperature was evaluated. The setup in Figure 2 was designed to work continuously with the ability to change the water or feed temperature. The feed temperature was controlled with a water bath a circulation heater and a frigidflow bath circulator, temperatures range from 4 to 40 °C. The water temperature was controlled with a PVC heat exchanger installed right before the water inlet, temperatures range from 0 to 40 °C.

Analysis of the diffusion dialyzer performance
The diffusion dialyzer performance was evaluated according to the method suggested by Tokuyama Soda Co., LTD (Tokuyama, 1991). The method implies the measurement of several variables: water and feed temperature, running time, stream volumes, stream acid and metal concentrations, and we added the recovery outlet level length. With these data we are able to calculate the following parameters: % of recovered acid, % of recovered normality, % of metal leakage, % of recovered acid volume, % of waste volume, overall mass transfer coefficient, recovery side hydrostatic pressure, waste side hydrostatic pressure, and hydrostatic pressure difference across the membrane. The first three parameters characterized the performance of the separation, while the rest of the parameters help us understand how it happened.

The recovered acid and waste volumes were measured directly with a 100 ml graduated cylinder. The flow rates were obtain dividing the volumes by the elapsed time. The percentage of recovery volume was calculated dividing the recovered acid volume over the total volume of the outlets (recovery + waste), and multiplying it by 100. The percentage of waste volume was calculated dividing the waste volume over the total volume, and multiplying by 100. The water and feed temperatures were measured with thermocouple digital thermometers installed at the water inlet and in the water bath, an assumption was made, that the feed will equilibrate with the bath temperature before it enters the dialyzer. The recovery outlet level was measured with the help of a ruler installed permanently in the dialyzer.

The acid concentration was evaluated by titration, 5 ml of the sample was diluted
in 95 ml of DI water. Titration was performed using a 9.8 M sodium hydroxide solution with addition of bromothimol blue solution (0.04%) as an indicator, the end point of the titration was assumed at a pH of 7 corresponding to a blue color. The volume of sodium hydroxide needed to obtain a change in color from yellow to blue was reported (Clesceri et al, 1989). Calculation of sample molarity was obtained according to equation (70)

$$M_s = \frac{(M_{NaOH} \cdot V_{NaOH})}{V_s} \qquad (70)$$

where "$M_s$" is the sample molarity, "$V_s$" is the sample volume (5 ml), "$M_{NaOH}$" is the molarity of the sodium hydroxide (9.8M), and "$V_{NaOH}$" is the volume of sodium hydroxide needed to titrate the sample. The normality was calculated by multiplying the molarity with the anion valence, in the case of nitric acid, the normality is equal to the molarity, and in the case of sulfuric acid the molarity is half the normality. The moles of acid, in any of the three streams (feed, waste, and recovery) were calculated by multiplying the stream volume with the stream molarity. The total number of acid moles could be obtained by the feed moles or by the addition of the waste acid moles and the recovered acid moles. The percentage of recovered acid was calculated by dividing the recovered acid moles over the total number of acid moles, multiplied by 100. The percentage of recovered normality was calculated by dividing the normality of the recovered acid by the normality of the feed, multiplied by 100.

The metal concentration was obtained using an atomic absorption spectrophotometer (Perkin-Elmer 5000, AA), the recovered acid samples were
diluted 1:100 and 1:500 for iron and copper respectively, the waste samples were diluted 1:1000 for iron and copper. The dilution was needed so that the samples could be in the desired absorbance range. The metal concentration in the solutions was obtained by multiplying the AA value times the dilution factor. The amount of metal in every stream was calculated by multiplying the metal concentration with the stream volume. The total amount of metal in a run is equal to the amount of metal in the feed or the addition of the amount of metal in the recovered acid plus the waste. The percentage of metal leakage was calculated dividing the amount of metal in the recovered acid by the total amount of metal, multiplied by 100 (Clesceri et al, 1989).

The overall mass transfer coefficient was calculated according to equation (71), where "K" is the overall mass transfer coefficient, "A" is the effective membrane area, "W" is the weight of solute transferred across the membrane per unit of time, "ΔC<sub>in</sub>" is the in-mean transmembrane concentration difference for a continuous dialysis system, based on in-out let values.

$$K = \frac{W}{A \Delta C_{in}} \quad (71)$$

The following three chapters are written in a manuscript style, and represent different variables tested on diffusion dialysis. The experimental section on each of the following three chapters was written on a publication style form, therefore we thought an extended materials and methods would be beneficial to the dissertation as a whole.
Typical data sheet are as follow:

<table><caption>Table 1. Diffusion dialysis run data sheet</caption><thead><tr><th>#</th><th>TIME</th><th>H₂O<br>TEMP (C)</th><th>FEED<br>TEMP (C)</th><th>REC-<br>OUTLET (cm)</th><th>FEED<br>VOL (ml)</th><th>WAS<br>VOL (ml)</th><th>REC<br>VOL (ml)</th></tr></thead><tbody><tr><td>1</td><td>2:30</td><td>40</td><td>25</td><td>41.5</td><td>50</td><td>36</td><td>46</td></tr><tr><td>2</td><td>3:30</td><td>40</td><td>25</td><td>41.5</td><td>50</td><td>42</td><td>57</td></tr><tr><td>3</td><td>4:30</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>Table 2. Volume calculations data sheet.</caption><thead><tr><th>#</th><th>H₂O<br>VOL (ml)</th><th>TOT<br>VOL (ml)</th><th>% REC<br>VOL (ml)</th><th>% WAS<br>VOL (ml)</th></tr></thead><tbody><tr><td>1</td><td>32.0</td><td>82.0</td><td>56</td><td>44</td></tr><tr><td>2</td><td>49.0</td><td>99.0</td><td>58</td><td>42</td></tr><tr><td>3</td><td>55.0</td><td>106</td><td></td><td></td></tr></tbody></table>
<table><caption>Table 3. Titration data sheet.</caption><thead><tr><th>#</th><th>REC<br>NaOH<br>(ml)</th><th>REC<br>N</th><th>REC<br>MOLES</th><th>WAS<br>NaOH<br>(ml)</th><th>WAS<br>N</th><th>WAS<br>MOLES</th><th>TOT<br>MOLES</th><th>%<br>Rec</th><th>%<br>N</th></tr></thead><tbody><tr><td>1</td><td>4.4</td><td>8.56</td><td>0.3964</td><td>0.95</td><td>1.84</td><td>0.0671</td><td>0.4635</td><td>85.5</td><td>85.6</td></tr><tr><td>2</td><td>3.95</td><td>7.68</td><td>0.442</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>Table 4. Atomic absorption data sheet.</caption><thead><tr><th>#</th><th>REC<br>(ppm)</th><th>REC<br>(mg)</th><th>WAS<br>(ppm)</th><th>WAS<br>(mg)</th><th>TOT<br>(mg)</th><th>FEED<br>(ppm)</th><th>LEAK<br>%</th></tr></thead><tbody><tr><td>1</td><td>12,520</td><td>579</td><td>35,840</td><td>1,300</td><td>1,880</td><td>37,613</td><td>30.8</td></tr><tr><td>2</td><td>11,220</td><td>645</td><td>29,870</td><td>1,254</td><td>1,899</td><td>37,993</td><td>33.9</td></tr></tbody></table>
<table><caption>Table 5. Overall mass transfer coefficient data sheet.</caption><thead><tr><th>#</th><th>TEMP<br>(°C)</th><th>1/TEMP<br>(°C)</th><th>W<br>(g/hr)</th><th>ΔCln<br>(N)</th><th>ΔCln<br>(g/cm³)</th><th>U<br>(cm/hr)</th></tr></thead><tbody><tr><td>1</td><td>17</td><td>0.003448</td><td>10.32</td><td>1.00</td><td>0.04904</td><td>0.4211</td></tr><tr><td>2</td><td>25</td><td>0.003555</td><td>10.44</td><td>0.836</td><td>0.04099</td><td>0.5096</td></tr></tbody></table>

<table><caption>Table 6. Hydrostatic pressure data sheet.</caption><thead><tr><th>#</th><th>ρ<sub>rec</sub><br>(kg/m3)</th><th>P<sub>rec</sub><br>(Pa)</th><th>ρ<sub>was</sub><br>(kg/m3)</th><th>P<sub>was</sub><br>(Pa)</th><th>ΔP<br>(Pa)</th></tr></thead><tbody><tr><td>1</td><td>1006</td><td>3972</td><td>1000</td><td>4068</td><td>-96</td></tr><tr><td>2</td><td>1075</td><td>4182</td><td>1000</td><td>4068</td><td>114</td></tr></tbody></table>

REFERENCES

Clesceri, S.L., Greenberg, E.A. and Trussell, R.R., **Standard Methods. For the**
**Examination of Water and Wastewater**, American Public Health Association,
Washington DC 1989, 2-31,3-13.

Tokuyama Soda CO., LTD., **Instruction Manual for Diffusion Dialysis Unit Model**
**TSD-2**, Tokuyama Japan 1991.
<figure><img src="image_3.png" /><figcaption>**Figure 1**
Diffusion Dialysis Equipment</figcaption></figure>
<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /></figure>
# Chapter III
TEMPERATURE EFFECT ON DIFFUSION DIALYSIS PERFORMANCE
## SUMMARY

Diffusion dialysis separations are known to be slow rate process compared to other, pressure driven, filtration technologies. As in all diffusion dependent process, temperature plays an important role in diffusion dialysis. The present paper deals with the effect of system temperature and temperature differences between water and feed. Diffusion dialysis performance depends on the amount and concentration of acid that has being separated. The acid recovery and concentration are determined by the anion and water flux, which are affected in turn by the temperature of the system. The experiments were divided into three parts: 1.) Temperature effect on diffusion dialysis of pure acid solution (nitric and sulfuric acid). 2.) Temperature effect on diffusion dialysis of acid/metal solutions (nitric/copper and sulfuric/iron). 3.) The effect of temperature differences between water and feed sides on diffusion dialysis of acid/metal solutions. A continuous countercurrent mono-membrane system, with a anion-exchange membrane, was used to perform the diffusion dialysis experiments. The temperature variation was between 0 and 40°C. The performance was evaluated based on the percentage of recovered acid, percentage of recovered acid normality, and percentage of metal leakage. Increasing temperatures increased the osmosis producing higher recovered acid concentrations and lower acid recovery. The presence of a metal increased acid recovery as we expected from Donnan equilibrium. High hydrated ion volume and low system temperatures correlated well with low overall mass transfer coefficient. High dissociation constants correlated well with lower metal leakage. We weren't able to find a significant difference in diffusion
dialysis performance by increasing the water temperature relative to the temperature of the system.

## INTRODUCTION

Diffusion dialysis (DD) is a membrane technology process that provides an alternative to the traditional way (neutralization) of treating spent acid baths (Masayuki, 1992; Nakamura, 1983). DD is able to separate acid (anion) from metal (cation) by the use of an anion-exchange membrane, as illustrated in Figure 1. DD separation is known to be slow rate process compared to other, pressure driven, membrane separation technologies. Several factors are known to affect diffusion dialysis separation rate, some of which are: concentration difference across the membrane, hydraulic flow rates, hydrostatic pressure difference across the membrane, and temperature (Bailey, 1990). As in all diffusion dependent process, temperature plays an important role in diffusion dialysis. Increasing the temperature of a diffusion dialysis system, increases the diffusion rate (flux) of the separation by means of higher molecular kinetics and lower solvent viscosity. Recently it has been found that, in countercurrent parallel diffusion dialysis systems, increasing the temperature of the water above that of the feed solution, appears to increase the rate of diffusion into the water side, due to the increased thermal energy of the system (Bailey, 1990). Another reason that has been proposed is, the formation of a laminar flow in the water stream, preventing convective flow in the compartment from density
differences. This way the increasing solute concentration, from top to bottom along the recovery channel (Figure 1), is maintained. Ordinary tap water is usually saturated with air, changes in temperature and composition may cause the formation of bubbles within the frames, reducing the effective transfer area. For this reason, de-aeration of water by purging with hydrogen or by boiling the water is recommended. High temperature water has lower air solubility, which reduces the formation of bubbles in the frame compartment (Sato et al, 1985; Tuwiner, 1962). This paper will deal with the effect of temperature differences, between water and feed in countercurrent parallel diffusion dialysis, testing some of the statements mentioned above.

Since diffusion dialysis membranes are made out of a polymeric matrix to which ionizable groups are covalently bound, temperature is a limiting factor (Mears, 1983). Temperatures higher than the membrane's limit, affect the polymeric crosslinking modifying the permeability of the membrane. For most diffusion dialysis membranes the temperature allowance is between 0° - 40°C (Tokuyama Soda, 1991).

Heating up the water or feed is an energy expending process, which in an industrial situation has to be minimized. For this reason and the temperature statements mentioned before, a deep understanding of the temperature effect over diffusion dialysis performance is a worth while investigation.

If we start with the extended form of the Nernst-Planck flux equation (equation 1) for diffusion dialysis, we can see that the flux is divided into two main sections; the convective flux "$C_i v$", and the diffusion flux "$D_i$ [..........]" (Schlögl, 1966).
$$J_i = C_i v^* - D_i \left[ \frac{dC_i}{dx} + C_i \frac{d \ln \gamma_i}{dx} + z_i \frac{F}{RT} C_i \frac{dE}{dx} + \frac{C_i}{RT} \left( V_i - \frac{M_i}{M_w} V_w \right) \frac{dp}{dx} \right] \quad (1)$$

In many dialysis applications where the convective flux is negligible (no hydrostatic pressure difference across the membrane), the diffusion flux plays the major role in the separation. In these cases, the diffusivity "$D_i$" contributes greatly to the separation performance. The mass diffusivity for a binary system is a function of temperature, pressure, and composition (Bird et al, 1960). If we assume the kinetic theory of gases applies to diluted solutions, the diffusivity can be expressed as

$$D = \frac{1}{3} \lambda \bar{u} \quad (2)$$

where "$\lambda$" is the mean free path, and "$\bar{u}$" is the molecule mean speed. The mean speed "$\bar{u}$" is calculated from a Maxwell distribution to obtain

$$\bar{u} = \left( \frac{8kT}{\pi m} \right)^{\frac{1}{2}} \quad (3)$$

where "$k$" is the Boltzman's constant, "$m$" is the molecular mass, and "$T$" is the temperature. Substituting equation (3) into equation (2), we can see that the diffusivity "$D_i$" is directly proportional to the temperature. Other relationships for diffusivity arise when the extended Nernst-Planck equation is simplified to Fick's first law of diffusion

$$J_i = -D_i \frac{dc_i}{dx} \quad (4)$$

Fick's first law can be rearranged into the Einsten relation between the diffusivity and
the ionic mobility

$$D_i = \frac{uRT}{zF} \qquad (5)$$

where "u" is the ion mobility, "R" is the gas constant, "z" is the ion charge, and "F" is the Faraday's constant. Another relationship is the Stokes-Einstein equation (6), relating the diffusivity with the frictional constant "f".

$$D_i = \frac{kT}{f} \qquad (6)$$

Unlike the gases, the diffusivity of liquids varies appreciably with concentration. Estimates of diffusivity in liquids without experimental data can not be made with anything like the accuracy for gases (Treybal, 1980). Since diffusivity relationships are directly proportional to temperature (increasing temperature increases diffusivity), molecules in a hot environment diffuse faster than those in a cool environment, for a given concentration gradient (Atkins, 1990).

Water transport is also affected by temperature, when differences in composition exist and the membrane is not freely permeable to all solutes (eg. ion-exchange membrane), the flux is affected by osmotic pressure differences across the membrane. Osmosis is defined as the movement of water from a diluted solution into a concentrated solution. The osmotic pressure can be calculated from Van't Hoff law as:

$$\Delta\Pi = RT \Delta C_i \qquad (6a) \quad (\text{Bungay et al, 1986})$$

as we can see from equation 6a, the osmotic pressure is proportional to temperature.
Therefore, as temperature increases, the osmotic pressure increases and so does the water flux. Another mechanism in which water can be transported across a membrane is by pressure difference. Diffusion dialysis systems do not use any applied pressure, but, in some situations hydrostatic pressure difference may arise across the membrane. In these situations the water transport is subjected to a "ΔP" that influence the water flux. The equation for water flux (equation 6b) "Jv" becomes the sum of the hydrostatic pressure and osmotic pressure as:

$$J_v = L_p (\Delta P - RT \Delta C_i) \quad (6b)$$

where "Lp" is the mechanical coefficient of filtration or water permeability (Lakshminarayanaiah, 1969).

Most reaction rates increase as the temperature is raised. An empirical observation is that many reactions have rate constants that follow the Arrhenius equation, equation (7)

$$\ln k = \ln A - \frac{E_a}{RT} \quad (7)$$

where "A" is the pre-exponential factor and "Ea" is the activation energy. The activation energy can be obtained by plotting "1/T" vs "ln k", where the slope of the line is equal "-Ea/R". The activation energy is the minimum energy that the reactants must have in order to form the products (Atkins, 1990; Moore, 1962). Mass transfer in diffusion dialysis can be model as follows:

$$J_i = K A C_m \quad (8)$$

where "K" is the overall mass transfer coefficient, "A" is the effective membrane
area, and "Cₙ" is the logarithmic concentration mean difference (Nakamura et al, 1983; Klein et al, 1987). This equation resembles a reaction rate equation, where "Jᵢ" is the product, "Cₙ" is the reactant, and "K" is the reaction rate constant. Using this analogy, the effect of temperature in diffusion dialysis can be analyzed with the use of the Arrhenius equation. If the diffusivity increases rapidly with rise in temperature, it's an evidence that the activation energy is considerable (Tuwiner, 1962). The activation energy would be the minimum amount of energy that the solute needs in order to cross the membrane.

# MATERIALS AND METHODS

## Acid Solutions

Nitric and sulfuric acid solutions were prepared according to the following acid and metal concentrations:

1) Sulfuric Acid 5 N / Iron 0.0 ppm

2) Sulfuric Acid 5 N / Iron 16,500 ppm

3) Nitric Acid 10 N / Copper 0.0 ppm

4) Nitric Acid 10 N / Copper 30,000 ppm

Acid solutions were prepared using analytical reagents from Fisher Scientific.

## Diffusion Dialysis Set Up

Diffusion dialysis separations were performed using the Tokuyama Soda Co., LTD., model TSD-5-1, with an anion-exchange membrane Neosepta model AFN. The
instrument was run according to the arrangement in Figure 2. All of the runs were
performed with a minimum of two repetitions. The water temperature was adjusted
with the heat exchanger shown on Figure 2. The system temperature was adjusted
using the water bath/circulation heater shown in Figure 2. Samples were collected for
1 hr after a minimum running time of 5 h. Feed and water flow rates were held
constant at 1.0 l/m²-hr. The recovery outlet level (shown in Figure 2 as Hrec) was
held constant at 41.5 cm.

Evaluation of Diffusion Dialysis

The performance of the separation was evaluated according to the procedures
recommended by Tokuyama Soda Co., LTD., also used by Sridhar and Subramaniam
(1989). The activation energy was calculated with the use of the Arrhenius equation
(equation 7). The parameters measured were: acid (by titration) and metal (by
atomic absorption spectroscopy) concentration in the feed, recovered acid, and waste
streams (Clesceri et al, 1989). Water, feed, recovered acid, and waste flow rates
were also determined by measuring the respective volumes after one hour of running.
The hydrostatic pressure is calculated according to the formula "ρgh", where "ρ" is
the recovery or waste density, "g" is the gravitational force (9.8 m/sec²) and "h" is
the fluid column length (41.5 cm). The calculated parameters were: hydrostatic
pressure difference across the membrane "ΔhP" (recovery hydrostatic pressure "hPrec"
- waste hydrostatic pressure "hpwas"), percentage of volume changes in recovered acid
and waste, percentage of recovered acid, normality, and metal leakage, and overall
mass transfer coefficient.
## Temperature Variation

Experiments were divided into two different variables: water temperature and feed temperature. The water temperature was controlled using a PVC heat exchanger located at the entrance of the water stream (shown in Figure 2), a thermocouple was placed between the dialyzer and the heat exchanger to monitor the temperature. The feed temperature was controlled using a water bath, a circulator heater (shown in Figure 2) and a Frigidflow bath circulator. The feed was assumed to have the bath temperature before it entered the dialyzer, a thermocouple was placed before the entrance to monitor feed temperature. Temperature variation in both cases range from 3° - 40°C.

## RESULTS AND DISCUSSION

### Temperature Effect on Sulfuric and Nitric acid Diffusion Dialysis

Figures 3 and 3A show the effect of temperature on hydrostatic pressure, for sulfuric and nitric acid respectively. The recovery side hydrostatic pressure increases linearly ($r = 0.98$ sulfuric acid, $r = 0.97$ nitric acid) with temperature. The hydrostatic pressure increase is due to a reciprocal increase in the density of the recovery side fluid. The density of the recovery is affected by two phenomena; diffusion of anions (sulfates and nitrates) from the feed into the recovery, and diffusion of water (osmosis) from the recovery into the feed. As diffusion of anions and/or osmosis increases the density of the recovery stream increases. The waste side
hydrostatic pressure as well as the density remains constant ($P = 4175 \pm 14$ pa sulfuric acid, $P = 4189 \pm 10$ pa nitric acid) as temperature increases. Since the density in the waste side depends on the same two phenomena (anion diffusion and osmosis) mentioned above, that must be a balance between the anion diffusion and osmosis at every temperature, in order to maintain the waste density constant. As a result of an increase in the recovery hydrostatic pressure and a constant waste hydrostatic pressure, a linear increase in hydrostatic pressure difference ($r = 0.95$ sulfuric acid, $r = 0.94$ nitric acid) across the membrane develops with increasing temperature, as Figures 3 and 3A show. Figures 4 and 4A show the effect of temperature on the recovered acid and waste volumes, for sulfuric and nitric acid respectively. Recovered acid volume (% Recovery Vol) decreases linearly ($r = 0.95$ sulfuric acid, $r = 0.86$ nitric acid), while the waste volume (% Waste Vol) increases linearly ($r = 0.95$ sulfuric acid, $r = 0.86$ nitric acid), as the temperature rises. This phenomena, as I mentioned before, is called "osmosis", where the water diffuses from a diluted compartment into a concentrated one. Countercurrent parallel dialyzer has the advantage of maintaining a concentration gradient (anion concentration at feed side higher than the recovery side) at any point along the membrane, this, along with the hydrostatic pressure difference, favors the diffusion of water from the recovery side into the waste side calling it "osmosis". Since osmosis is considered part of a diffusion process, temperature affects osmosis based on an increasing molecular kinetics. Figures 5 and 5A show the effect of temperature over the percentage of recovered acid normality, for sulfuric and nitric acid respectively. The percentage of
recovered acid (Recovery %) decreases as temperature increases, while the percentage of normality (Normality %) increases linearly (r = 0.98 sulfuric acid, r = 0.97 nitric acid) as temperature increases. The normality of an acid solution is proportional to its density. The density and the normality of a recovered acid solution depend on the osmosis and anion diffusion. Therefore, the increasing normality with temperature has to be related to both or at least one. Since osmosis increases with temperature, as shown in Figures 4 and 4A, the concentration of anions (acid) in the recovery side rises, increasing the normality and the density of the recovery solution. The second alternative is the diffusion of anions into the recovery side, as shown in Figures 5 and 5A, the percentage of recovered acid decreases with increasing temperature, even though the overall mass transfer coefficient increases. For this reason only the osmosis is responsible for the increase in normality with increasing temperatures.

The decreasing of anion diffusion could be related to the increasing flow resistant by the convective water flow in the opposite direction. It appears that the diffusion of nitric acid is less affected by the temperature compared to the sulfuric acid, as Figure 5 and 5A show. This difference, again could be related to the differences in osmosis, sulfuric acid shows higher osmosis than nitric acid (Figure 4 and 4A). Figure 6 and 6A show the Arrhenius Plot for sulfuric and nitric acid respectively. The activation energy was obtained based on the slope of the plots. The activation energy "Ea" for sulfuric and nitric acid are 4.45 KJ/mol and 6.37 KJ/mol respectively. Based on these values, nitric acid appears to have a stronger temperature dependence compare to sulfuric acid. Therefore, diffusion dialysis of nitric acid will be more affected by
temperature than sulfuric acid. The fact that the activation energy of nitric acid is greater than the one for sulfuric acid, does not mean that the sulfuric acid will diffuse faster than the nitric. As we can see from Figure 6 and 6A, the overall mass transfer coefficient "K" for nitric acid is greater than the one for sulfuric acid, within the temperature range. Therefore, diffusion dialysis of nitric acid has higher recovery percentage than sulfuric acid, as Figure 5 and 5A show. Another factor called the pre-exponential factor "A", can be obtained from the Arrhenius plot. This factor is also called the ideal overall mass transfer coefficient "K₀", which is equal to "K" when the volume of the solute being transferred is zero (Tuwiner, 1962). This factor "K°" depends on the molecular volume of the ion that is being transported across the membrane. From Figure 4 and 4A the "K°" values for sulfuric and nitric acid are K° = 1.92 cm/hr and K₀ = 8.25 cm/hr respectively. This values correlates well with the hydrated ion volume "v_H". Mandal and coworkers (1973) found the following volumes for hydrated sulfate "SO₄" and nitrates "NO₃", v_H = 137.45 cm³ and v_H = 94.92 cm³ respectively. This could be the main reason why nitric acid separates better in diffusion dialysis than sulfuric acid.

## Temperature Effect on Sulfuric acid / Ferrous Sulfate and Nitric acid / Copper Nitrate
Diffusion Dialysis

Figure 7 and 7A show the effect of temperature on hydrostatic pressure, for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate solutions respectively.
As I mentioned on the previous section, the recovery side hydrostatic pressure increases linearly (r = 0.96 sulfuric acid / ferrous sulfate, r = 0.95 nitric acid /
copper nitrate) with temperature. The waste side hydrostatic pressure remains constant ($P = 4132 \pm 6$ pa sulfuric acid / ferrous sulfate, $P = 4163 \pm 16$ pa nitric acid / copper nitrate). As a result of an increasing recovery side hydrostatic pressure and a constant waste hydrostatic pressure with increasing temperature, a linear increase in ($r = 0.95$ sulfuric acid / ferrous sulfate, $r = 0.93$ nitric acid / copper nitrate) hydrostatic pressure difference across the membrane develops with increasing temperature. In the case of sulfuric acid, the recovery side hydrostatic pressure remains the same with or without ferrous sulfate (equation 9 and equation 20) but the waste hydrostatic pressure changes from $P = 4175 \pm 14$ pa in sulfuric acid to $P = 4132 \pm 6$ pa in sulfuric acid / ferrous sulfate, these leads to a greater hydrostatic pressure difference in the presence of ferrous sulfate. In the case of nitric acid, the recovery side hydrostatic pressure drops and the waste hydrostatic pressure increases with the presence of copper nitrate, producing a smaller hydrostatic pressure difference across the membrane. Figure 8 and 8A show the effect of temperature on the recovered acid and waste volumes, for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate respectively. The percentage of recovery volume decreases linearly ($r = 0.92$ sulfuric acid / ferrous sulfate, $r = 0.95$ nitric acid / copper nitrate), while the percentage of waste volume decreases linearly ($r = 0.92$ sulfuric acid / ferrous sulfate / nitric acid / copper nitrate), as the temperature increases. The volume changes in nitric acid / copper nitrate (Figure 8A) are identical to the changes in pure nitric acid (Figure 4A), therefore, the difference in osmotic pressure, between the acid metal solution and the pure acid solution, is negligible. Copper leakage may
be the reason for the small osmosis effect in the volumes. In the case of sulfuric acid / ferrous sulfate the 50/50 volume equilibrium is reached at a higher temperature (Figure 8, T = 32°C) compared to the pure sulfuric acid (Figure 4, T = 23°C).

Again the osmotic pressure is negligible compared to the water of hydration by the transport of sulfate ions. Figure 9 and 9A show the effect of temperature over the percentage of recovery, normality, and metal leakage, for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate respectively. The percentage of recovery is constant at % Rec = 86.13 ± 0.53 and 88.15 ± 1.53 for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate respectively, the percentage of normality increases linearly (r = 0.97 sulfuric acid / ferrous sulfate, r = 0.95 nitric acid / copper nitrate), and the percentage of metal leakage is constant at % leakage = 10.74 ± 0.97 for sulfuric acid / ferrous sulfate, and % leakage = 31.07 ± 1.91 for nitric acid / copper nitrate. In both cases the presence of the metal increases the anion diffusion to maintain the percentage of recovery constant at all temperatures. This can be explained by the use of the Donnan equilibrium and the electroneutrality requirement (Lake and Melshedaer, 1978; Cwirko and Carbonell, 1990; Kessler and Klein, 1992). The Donnan equilibrium for diffusion dialysis of sulfuric acid / ferrous sulfate can be expressed as fallow:

$$ \left[ \frac{aSO_4 R}{aSO_4 W} \right]^{\frac{1}{z_s}} = \left[ \frac{aH R}{aH W} \right]^{\frac{1}{z_H}} \quad (9) $$

where "z_s" and "z_H" are the ion valence for sulfate and hydrogen respectively, "aSO_4 R" and "aSO_4 W" are the activities of sulfate in the recovery and the waste side
respectively, "aH R" and "aH W" are the activities of hydrogen in the recovery and waste side respectively. The activity of sulfate in the recovery side can be isolated from equation (9), substituting the "z" values equation (9) turns into

$$aSO_4 \cdot R = \left( \left[ \frac{aH \cdot R}{aH \cdot W} \right] [aSO_4 \cdot W]^{aH} \right)^{\frac{1}{2}} \quad (10)$$

The activity of hydrogen in the waste side is always restricted by the electroneutrality requirement. When no metal is present in the solution, any remaining sulfates in the waste side have to be neutralized by hydrogens. But, when metal is present in the solution, and it is excluded from the recovery side by the membrane, the activity of hydrogen in the waste side is reduced to a much smaller number, since the metal covers the function of neutralizing the remaining sulfates. In this situation the lower activity of the hydrogen in the waste side, increases the activity of the sulfate in the recovery side, as equation (10) shows. The same explanation applies to the nitric acid / copper nitrate solution, but, the effect is less obvious because the exclusion of copper is not as strong as with iron. The percentage of normality as we mentioned before, depends on the osmosis and anion diffusion. In the case of nitric acid Figures 4A and 8A show almost identical osmosis with or with out the presence of copper.

Therefore, the normality will be solely affected by the difference in nitric acid diffusion with and without copper. As we seen in Figures 5A and 9A the recovery percentage increased slightly with the presence of copper, therefore the normality should increase slightly as well. Figures 5A and 9A show exactly this and it can be verified by the higher slope in the normality linear regression (equation 35) for the
nitric acid / copper nitrate solution, compared with equation (21). In the case of sulfuric acid, the normality percentage has a lower slope in the presence of iron, Figures 5 and 9. This could be related to the lower osmosis taking place in the iron acid solution as Figures 4 and 8 show. Therefore, osmosis is a more important factor when it comes to acid concentration (normality). The metal leakage can be seen in Figures 9 and 9A for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate respectively. The percentage of leakage remains constant as the temperature increases with a higher value for copper (% leakage = 31.07 ± 1.91) than iron (% leakage = 10.74 ± 0.97). The difference in metal leakage can be related to the size of the hydrated metal and the equilibrium constant between the salt and the ion state. The higher the size of the hydrated metal the lower the leakage. Lower dissociation constants "K" produce higher leakage, since the salt metal form has no charge and lower hydrated volume, compared to the ion state. Iron ions present higher hydrated volume ($v_HFe = 198 \text{ cm}^3\text{/mol}$) than copper ($v_HCu = 185.7 \text{ cm}^3\text{/mol}$), and ferrous sulfate has a higher dissociation constant ($K_{FeSO_4} = 0.04$) than copper nitrate ($K_{Cu(NO_3)_2} = -16.19$) (Horvath, 1985; Bjerrum, 1958). This two properties correlate well with the higher percentage of leakage by the copper. Figures 10 and 10A show the Arrhenius plots for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate respectively. The activation energies for the acid / metal solutions are higher than the ones for the pure acid solutions. The overall mass transfer coefficient for sulfates increases with the presence of iron while the overall mass transfer coefficient for nitrate decreases with the presence of copper. This phenomena could be related to the
metal leakage capacity, since I have shown that the capacity of a membrane to exclude a metal affects the diffusion of the anion by Donnan equilibrium. The same pattern as in Figures 4 and 4A is observed again, the activation energy for the nitric acid / copper nitrate solution (Ea = 19.54 KJ/mol) is higher than the activation energy for the sulfuric acid / ferrous sulfate solution (Ea = 10.80 KJ/mol).

Therefore, in diffusion dialysis, the separation of nitric acid is more likely to be affected by temperature than the separation of sulfuric acid.

## Water Temperature Effect on Sulfuric and Nitric acid Diffusion Dialysis

Figures 11 and 11A show the effect of water temperature on hydrostatic pressure, for sulfuric acid / ferrous sulfate and nitric acid / copper nitrate respectively. The recovery hydrostatic pressure, waste hydrostatic pressure, and pressure difference are constant as the water temperature increases. Therefore, the acid concentration in the recovery and the waste are independent of the water temperature. This is shown in Figures 13 and 13A by a straight line for percentage of normality. Figures 12 and 12A show the effect of water temperature over the recovery and waste volume percentage. The percentage of recovery volume as well as the percentage of waste volume remain constant as the temperature of the water increases, % Rec Vol = 55.00 ± 2.03 for nitric acid / copper nitrate, and % Rec Vol = 51.33 ± 1.86 for sulfuric acid / ferrous sulfate, there is a very small tendency to increase the volume of recovery as the temperature increases, this phenomena is called thermoosmosis, which is the diffusion of water from the cold side into the hot side (Lakshminarayanaiah, 1969). There is no significant change in percentage of
recovery, normality or metal leakage as the temperature of the water increases. It is mentioned by some authors that the water temperature needs to be at a higher temperature than the feed, to prevent convection, maintain the water inlet in a laminar flow, or simply to increase the acid recovery (Bailey, 1989). As I mentioned before, I have found no significant difference in acid recovery, acid concentration or metal leakage by increasing the temperature of the water. We found that the density gradient along the recovery compartment, for a (10 N HNO<sub>3</sub>/30,000 ppm Cu) nitric acid / copper nitrate diffusion dialysis, has a profile as Figure 12 shows. In countercurrent diffusion dialysis, water flows from top to bottom, and feed flows from bottom to top (Graver, 1993; Ondrick, 1993; Tokuyama, 1985). The maximum achievable density for water is ρ = 999.972 Kg/m<sup>3</sup> at a temperature of 4°C, as Figure 14 shows the density of the recovery side at the top of the dialyzer is much greater than the maximum water density. Therefore it is impossible to have any convective flow from density differences. We also calculated the Reynolds number for the recovery side fluid at water and feed flow rates of 1.25 l/hr-m<sup>2</sup>, using an equivalent hydraulic diameter for our rectangular duct (membrane compartment) (Re = 0.30), which is well under the limit for laminar flow (White, 1986). Any possible increase in velocity by convection will not create a turbulent flow. I do agree that by increasing the temperature of "both" feed and water, the molecular kinetics increase, from which we have found that the acid concentration benefits by the osmosis effect, but the overall acid recovery is not affected.
CONCLUSIONS

The hydrostatic pressure difference across the membrane increases linearly with temperature. The main reason for this increase is an increase in the recovery hydrostatic pressure from higher densities. The recovered acid volume decreases linearly with temperature because of pressure and concentration differences across the membrane. Water diffuses from the recovery side into the waste side by pressure gradient and osmosis. The reduction in volume in the recovery increases the acid concentration linearly (higher normalities), but at the same time the anion (acid) diffusion decreases, probably from an increase resistance from the water flow. The overall mass transfer coefficient "K" increases with temperature, but the resistance from the water flow over-power the increase in "K", reducing the overall acid flux.

It is expected that reducing the convective flow of water into the waste side by reducing the hydrostatic pressure difference, will increase the anion flux into the recovery side. The separation of nitric acid is more affected by temperature than sulfuric, since nitric acid shows higher activation energies, however, the mass transfer coefficient for nitric acid is higher than the one for sulfuric. Therefore, the separation of nitric acid yields better efficiency. The differences in over all mass transfer coefficient correlated well with the hydrated ion volume, higher hydrated ion volumes produce lower mass transfer coefficients, assuming equal charge. The osmotic pressure difference, between metal and un-metal solutions, has negligible effect over the outlet volumes. The metal leakage remained constant with increasing temperature. Copper shows higher leakage than iron, these values correlated well
with a higher hydrated volume and higher disassociation constant for the iron. Anion diffusion increased in the presence of metal ions, as required Donnan equilibrium and electroneutrality. The effect of water temperature over the acid recovery was negligible, convection flow by differences in density in the recovery side is nearly impossible, and in the remote event of a convection flow, the normal parameters for diffusion dialysis, provide a Reynolds number very well within the laminar range.

BIBLIOGRAPHY

Atkins, P.W., *Physical Chemistry*, Fourth ed., W.H. Freeman and Company, NY 1990, 737,761-769,792.

Bailey, D.E., Recovery and Reuse of Spent Acid Baths via Diffusion Dialysis, *Hazardous and Municipal Waste Minimization Conference, Providence RI* Feb. 7-8, 1989.

Bird, R.B., Stewart, W.E., and Lightfoot, N.E., *Transport Phenomena*, Jhon Wiley and Sons, NY 1960, 504-505.

Bjerrum, G.S., and Sillen, L.G., Stability Constants of Metal-ion Complexes, with Solubility Products of Inorganic Substances, Part II: Inorganic Ligands, *The Chemical Society, Burlington House 1958*, Special Publication No.7, 55,43,82-83.

Bungay, P.M., Lonsdale, H.K., and de Pinho M.N., *Synthetic Membranes: Science, Engineering and Applications*, ed. D. Reidel Publishing Company, Dordrecht 1986, 109-112.

Clesceri, S.L., Greenberg, E.A. and Trussell, R.R., *Standard Methods, For the Examination of Water and Wastewater*, American Public Health Association, Washington DC 1989, 2-13,3-13.

Cwirko, E.H. and Carbonell, R.G., A Theoretical Analysis of Donnan Dialysis Across Charged Porous Membranes, *Journal of Membrane Science*, 1990, 48, 155-179.
Graver Water, Division of The Graver Company, **Membrane Technologies**, Diffusion Dialysis, Union NJ 1993.

Horvath, A.L., **Handbook of Aqueous Electrolyte Solutions**, John Wiley and Sons, NY 1985, 344.

Kessler, S.B. and Klein, E., **Membrane Handbook**, ed. W.S. Winston Ho and K.K. Sirkar, Van Nostrand Reinhold, NY 1992, 161-215.

Klein, E., Ward, R.A. and Lacey, R.E., **Handbook of Separation Process Technology**, ed. R.W. Rousseau, John Wiley & Sons, NY 1987, 954-981.

Lake, M.A. and Melshedaer, S.S., Mass Transfer Characterization of Donnan Dialysis, *American Institute of Chemical Engineering Journal*, 1978, 24, 1, 130-137.

Lakshminarayanaiah, N., Transport Phenomena in Membranes, Academic Press, NY & London 1969, 58-361.

Mandal, P.K., Seal, B.K., and Basu, A.S., Z.Phys. Chem. Neue Folge, 1973, 87, 4-6

Masayuki, O., Uchida, M., and Ishibashi, T., Stainless Steel Pickling by Fixed Concentration HF/HNO₃ Mixtures Through Diffusion Dialysis Acid Reclamation, *Wire Journal International*, 1992, March, 43-48.

Mears, P., Synthetic Membranes: Science, Engineering and Applications, ed. P.M. Bungay, H.K. Lonsdale and M.N. de Pinho, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo 1986, 169-179.

Moore, W.J., Physical Chemistry, Third ed., Prentice-Hall INC., NJ 1962, 273-274.

Nakamura, I., Kawahara, T., and Utsunomiya, T., Use of Dialysis Techniques in Metal Finishing Process, New Materials and New Processes, 1983, 2, 166-175.

Ondrick, E., Use of Diffusion Dialysis to remove Metals from Nickel Plating Solutions at a Nickel Plating Job Shop, New England Water Environment Association Zero Discharge Conference, New Hampshire May 27, 1993.

Perry, R.H., and Green, D.W., Perry's Chemical Engineers' Handbook, 6th ed, McGraw-Hill, NY 1984, 17_18-17_26.
Tokuyama Soda Co., LTD, Reprint from Jitsumu Hyomen Gijutsu, *The Metal Finishing Society of Japan*, 1985, 1-8.

Tokuyama Soda CO., LTD., *Instruction Manual for Diffusion Dialysis Unit Model TSD-2*, Tokuyama Japan 1991.

Treybal, R.E., *Mass-Transfer Operations*, Third ed., McGraw-Hill, NY 1980, 34-36.

Sato, J., Onuma, M., Motomura, H. and Noma, Y., Recovery of Nitric Acid and Hydrofluoric Acid from the Pickling Solution by Diffusion Dialysis, Reprinted from Jitsumu Hyomen Gijutsu, *The Metal Finishing Society of Japan*, 1985.

Schlögl, R., *Berichte der Bunsengesellschaft fuer Physikalische Chemie*, 1966, 71, 755.

Sridhar, P., and Subramaniam, G., Recovery of Acid From Cation Exchange Resin Regeneration Waste by Diffusion Dialysis, *Journal of Membrane Science*, 1989, 45, 273-280.

Tuwiner, S.B., *Diffusion and Membrane Technology*, Reinhold Publishing Corporation, NY & London 1962, 38-60, 77-90, 137-151, 287-320.

Wallace, R.M., Concentration and Separation of Ions by Donnan Membrane Equilibrium, *Industrtial and Engineering Chemistry, Process Design and Development*, 1967, 6, 423-431.

White, F.M., *Fluid Mechanics*, Second ed., McGraw-Hill, NY 1986, 322.
<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /></figure>
<figure><img src="image_8.png" /><figcaption>Figure 3
Sulfuric acid 5 N, temperature effect on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$r = 0.98$

$$P_{\text{rec}} = 2.42T + 4354.29 \quad (11)$$

Waste Pressure Average and Standard Deviation

$$P_{\text{was}} = 4174.71 \pm 13.81$$

Pressure Difference Linear Regression

$r = 0.95$

$$\Delta P = 2.79T + 170.72 \quad (12)$$
<figure><img src="image_9.png" /><figcaption>Figure 3A
Nitric acid 10 N, temperature effect on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$r = 0.97$

$$P_{\text{rec}} = 2.77T + 4504.53 \quad (13)$$

Waste Pressure Average and Standard Deviation

$$P_{\text{was}} = 4188.71 \pm 10.21$$

Pressure Difference Linear Regression

$r = 0.94$

$$\Delta P = 2.71T + 318.36 \quad (14)$$
<figure><img src="image_10.png" /><figcaption>Figure 4
Sulfuric acid 5 N, temperature effect on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Linear Regression

$$r = 0.95$$
$$V_{\text{rec}} = -0.68T + 66.09 \quad (15)$$

Waste Volume Linear Regression

$$r = 0.95$$
$$V_{\text{was}} = 0.68T + 33.90 \quad (16)$$
<figure><img src="image_11.png" /><figcaption>Figure 4A
Nitric acid 10 N, temperature effect on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Linear Regression

$$r = -0.86$$
$$V_{\text{rec}} = -0.30T + 62.36 \quad (17)$$

Waste Volume Linear Regression

$$r = 0.86$$
$$V_{\text{was}} = 0.30T + 38.34 \quad (18)$$
<figure><img src="image_12.png" /><figcaption>Figure 5
Sulfuric acid 5 N, temperature effect on acid recovery and normality percentage</figcaption></figure>

Statistical Analysis:

Normality Percentage Linear Regression

$r = 0.98$

$$N = 0.53T + 63.96 \quad (19)$$
<figure><img src="image_13.png" /><figcaption>Figure 5A
Nitric acid 10 N, temperature effect on acid recovery and normality percentage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Linear Regression

$r = -0.60$
$\text{Rec} = -0.09T + 89.79 \quad (20)$

Normality Percentage Linear Regression

$r = 0.97$
$N = 0.33T + 89.79 \quad (21)$
<figure><img src="image_14.png" /><figcaption>**Figure 6**
Sulfuric acid 5 N, temperature effect on overall mass transfer coefficient</figcaption></figure>

Statistical Analysis:

Arrhenius Plot Linear Regression

$r = -0.88$

$$ \ln K = -535.68(1/T) + 0.6531 \quad (22) $$

$E_a = 4.45 \text{ KJ/mol}$

$K_o = 1.92 \text{ cm/hr}$
<figure><img src="image_15.png" /><figcaption>Figure 6A
Nitric acid 10 N, temperature effect on
overall mass transfer coefficient</figcaption></figure>

Statistical Analysis:

Arrhenius Plot Linear Regression

$r = -0.77$

$$ \ln K = -766.93(1/T) + 2.1109 \quad (23) $$

$E_a = 6.37 \text{ KJ/mol}$

$K_o = 8.25 \text{ cm/hr}$
<figure><img src="image_16.png" /><figcaption>Figure 7
Sulfuric acid 5 N, temperature effect on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$$r = 0.96$$
$$P_{\text{rec}} = 2.08T + 4372.64 \quad (24)$$

Waste Pressure Average and Standard Deviation

$$P_{\text{wgs}} = 4131.50 \pm 5.70 \quad (25)$$

Pressure Difference Linear Regression

$$r = 0.95$$
$$\Delta P = 2.68T + 224.45 \quad (26)$$
<figure><img src="image_17.png" /><figcaption>Figure 7A
Nitric acid 10 N, temperature effect on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$r = 0.95$

$$P_{\text{rec}} = 2.93T + 4454.97 \quad (27)$$

Waste Pressure Average and Standard Deviation

$$P_{\text{was}} = 4162.66 \pm 16.27 \quad (28)$$

Pressure Difference Linear Regression

$r = 0.93$

$$\Delta P = 2.10T + 309.94 \quad (29)$$
<figure><img src="image_18.png" /><figcaption>**Figure 8**
Sulfuric acid 5 N, temperature effect on recovery and waste volumes</figcaption></figure>

**Statistical Analysis:**

**Recovery Volume Linear Regression**

$$r = -0.92$$
$$V_{\text{rec}} = -0.31T + 59.87 \quad (30)$$

**Waste Volume Linear Regression**

$$r = 0.92$$
$$V_{\text{was}} = 0.31T + 40.12 \quad (31)$$
<figure><img src="image_19.png" /><figcaption>Figure 8A
Nitric acid 10 N, temperature effect on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Linear Regression

$$r = -0.95$$
$$V_{\text{rec}} = -0.25T + 62.87 \quad (32)$$

Waste Volume Linear Regression

$$r = 0.95$$
$$V_{\text{wars}} = 0.25T + 37.12 \quad (33)$$
<figure><img src="image_20.png" /><figcaption>Figure 9
Sulfuric acid 5 N, temperature effect on acid recovery, normality and Fe leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Average and Standard Deviation

$$ \text{Rec} = 86.13 \pm 0.53 $$

Normality Percentage Linear Regression

$$ r = 0.97 $$

$$ N = 0.45T + 67.95 \quad (34) $$

Metal Leakage Percentage Average and Standard Deviation

$$ \text{Leakage \%} = 10.74 \pm 0.97 $$
<figure><img src="image_21.png" /><figcaption>Figure 9A
Nitric acid 10 N, temperature effect on
acid recovery, normality and Cu leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Average and Standard Deviation

$$ \text{Rec} = 88.15 \pm 1.53 $$

Normality Percentage Linear Regression

$$ r = 0.95 $$

$$ N = 0.36T + 70.14 \quad (35) $$

Metal Leakage Percentage Average and Standard Deviation

$$ \text{Leakage \%} = 31.07 \pm 1.91 $$
<figure><img src="image_22.png" /><figcaption>Figure 10
Sulfuric acid 5 N, temperature effect on
overall mass transfer coefficient</figcaption></figure>

Statistical Analysis:

Arrhenius Plot Linear Regression

$r = -0.90$

$$ \ln K = -1299.14(1/T) + 3.60 \quad (36) $$

$E_a = 10.80 \text{ KJ/mol}$

$K_o = 36.59 \text{ cm/hr}$
<figure><img src="image_23.png" /><figcaption>Figure 10A
Nitric acid 10 N, temperature effect on
overall mass transfer coefficient</figcaption></figure>

Statistical Analysis:

Arrhenius Plot Linear Regression

$r = -0.97$

$$ \ln K = -2350.93(1/T) + 7.28 \quad (37) $$

$E_a = 19.54 \text{ KJ/mol}$

$K_o = 1463.19 \text{ cm/hr}$
<figure><img src="image_24.png" /><figcaption>Figure 11
Sulfuric acid 5 N, water temp. effect on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Average and Standard Deviation

$$P_{\text{rec}} = 4381.50 \pm 4.92$$

Waste Pressure Average and Standard Deviation

$$P_{\text{was}} = 4143.33 \pm 8.40$$

Pressure Difference Average and Standard Deviation

$$\Delta P = 237.33 \pm 7.99$$
<figure><img src="image_25.png" /><figcaption>Figure 11A
Nitric acid 10 N, water temp. effect on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Average and Standard Deviation

$$P_{\text{rec}} = 4536.00 \pm 22.42$$

Waste Pressure Average and Standard Deviation

$$P_{\text{was}} = 4161.64 \pm 9.00$$

Pressure Difference Average and Standard Deviation

$$\Delta P = 374.21 \pm 17.81$$
<figure><img src="image_26.png" /><figcaption>Figure 12
Sulfuric acid 5 N, water temp. effect on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Average and Standard Deviation

$$V_{\text{rec}} = 51.33 \pm 1.86$$

Waste Volume Average and Standard Deviation

$$V_{\text{wgs}} = 48.66 \pm 1.86$$
<figure><img src="image_27.png" /><figcaption>Figure 12A
Nitric acid 10 N, water temp. effect on
recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Average and Standard Deviation

$$V_{\text{rec}} = 55.00 \pm 2.03$$

Waste Volume Average and Standard Deviation

$$V_{\text{was}} = 45.00 \pm 2.03$$
<figure><img src="image_28.png" /><figcaption>Figure 13
Sulfuric acid 5 N, water temp. effect on
recovery, normality and iron leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Average and Standard Deviation
$Rec = 80.00 \pm 2.53$

Normality Percentage Average and Standard Deviation
$N = 73.00 \pm 1.09$

Metal Leakage Percentage Average and Standard Deviation
$Leakage \% = 8.26 \pm 0.58$
<figure><img src="image_29.png" /><figcaption>Figure 13A
Nitric acid 10 N, water temp. effect on
recovery, normality and copper leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Average and Standard Deviation

$$
\text{Rec} = 89.18 \pm 1.24
$$

Normality Percentage Average and Standard Deviation

$$
N = 79.86 \pm 2.68
$$

Metal Leakage Percentage Average and Standard Deviation

$$
\text{Leakage \%} = 32.31 \pm 3.69
$$
## Chapter IV
EFFECT OF ACID AND METAL CONCENTRATION
ON DIFFUSION DIALYSIS PERFORMANCE
SUMMARY

Recycling is one of the most effective ways of reducing industrial pollution. The use of mineral acids for a wide variety of processes is very common among industry, typical processes are copper etching with nitric acid from gold parts and pickling of steel parts with sulfuric acid. Diffusion dialysis has been introduced as an alternative to the neutralization treatment, with three major advantages: 1) recycle of acid, 2) less consumption of a neutralizing agent, and 3) less production of sludge. Optimization of crucial parameters like recovered acid and metal concentration is a must, for industry to apply the technology. In this study sulfuric acid / ferrous sulfate and nitric acid / copper nitrate solutions were treated with a anion diffusion dialysis system. The effect of acid and metal concentrations on percentage of recovered acid, recovered normality, and metal leakage were studied. At constant recovery outlet level, higher acid concentrations in the spent acid solution (feed) produces lower acid % recovery, % normality and, % leakage. The reduction in these parameters was correlated with an increase in the hydrostatic pressure difference across the membrane. Increasing iron concentration in the sulfuric acid feed, increases the % of recovered acid. The increase in acid flux is due to Donnan equilibrium. High iron rejection by the membrane and a 2:1 valence relationship between hydrogen and sulfates, cause the sulfate flux to increase by the Donnan equilibrium. The opposite effect was observed with copper and nitric acid. Higher copper concentration in the spent bath solution results in lower acid recovery. The reduction in the nitrate flux is attributed to a smaller effect by the Donnan equilibrium and an increase in osmosis.
INTRODUCTION

The reduction of process waste and the recycle of chemicals are the most
effective ways to reduce industrial pollution. The costs and liability associated with
the disposal of industrial waste, becomes greater every year. In addition, product
quality from consistent acid baths is a must for international competition. The use of
mineral acid for a wide range of processes is very common. Some of the typical acid
processes are; aluminum anodizing, pickling of secondary steel products, aluminum
foil etching, acid baths in metal electroplating, and in general any mineral acid
process which dissolves metals (Nakamura et al, 1983; Masayukiet al, 1992; Hikari et
al, 1987). When metals are acid treated in a bath, the concentration of the metal in
the bath will gradually build up and interfere with the process. Thus it is necessary
to discharge the bath once its efficiency becomes too low. In a typical chemical
treatment sequence, this waste acid must be neutralized before discharge. The
chemical process involves the use of alkalinies to obtain an adequate pH. Sludge
forms when the metal precipitates at the higher pH. Diffusion dialysis has been
introduced as an alternative to the traditional process. The process of diffusion
dialysis allows the company to recycle between 80 - 90 % of the acid from the spent
bath. Therefore, the amount of alkaline solution needed and sludge produced, is
reduced by 80 - 90 %. Diffusion dialysis is a membrane separation process, in which
solutes diffuse selectively from a high concentration solution (feed) into a low
concentration solution (water), through an ion exchange membrane (Figure 1). When
using an anion-exchange membrane, depleted acid solutions are able to transfer the
anions into the water side, but not the metal cation. In this way the acid is separated from the metals. Electroneutrality must be maintained in each ion-exchange membrane compartment. In acid diffusion dialysis, the electroneutrality is achieved by the movement of hydrogen ions along with the acid ions (anions). Hydrogen is able to cross the membrane against positive charge repulsion forces, because of its small size and ion charge (Bailey and Howard, 1992; Lynch, 1985; Fritsch, 1991). The movement of anions across the membrane is based mostly on chemical potential difference, but in some situations, hydrostatic pressure differences play an important role. Several factors have been found to affect the performance of diffusion dialysis, some of them are; temperature, feed and water flow rate, hydrostatic pressure difference, and acid and metal concentration (Bailey, 1989; Sato et al, 1985; Sridhar and Subramaniam, 1989). Diffusion dialysis performance is evaluated based on three different aspects of the separation; 1.) the percentage of recovered acid, 2.) the percentage of metal leakage, and 3.) the percentage of recovered acid normality (Tokuyama, 1991). The first two parameters are widely used in diffusion dialysis studies, but the percentage of recovered normality is a term that has not been reported previously. In any acid separation the concentration of the recovered acid with respect to the original solution is a very important factor. This information is represented by the percentage of recovered normality. We have chosen to determine the effect of acid and metal concentration on the behavior of diffusion dialysis performance.
# EXPERIMENTAL

## Acid Solutions

Acid solutions were prepared using analytical reagents from Fisher Scientific.
Nitric and sulfuric acid solutions were prepared according to the following acid and
metal concentrations:

1) Sulfuric acid 10 N

2) Sulfuric acid 10 N / Iron 25,000 ppm (Stock solution for iron variation)

3) Sulfuric acid 10 N / Iron 5,500 ppm (Stock solution for sulfuric acid variation)

4) Iron solution 5,500 ppm (as ferrous sulfate)

5) Nitric acid 10 N

6) Nitric acid 10 N / Copper 60,000 ppm (Stock solution for copper variation)

7) Nitric acid 10 N / 30,000 ppm (Stock solution for nitric acid variation)

8) Copper solution 30,000 ppm (as copper nitrate)

## Diffusion Dialysis Setup

Diffusion dialysis separations were performed using the Tokuyama Soda Co.,
LTD. equipment, model TSD-5-1 with an anion-exchange membrane Neosepta AFN.
All of the runs were performed with a minimum of two repetitions. Determination of
the variations in acid or metal concentration were performed continuously, using the
arrangement in Figure 2. Samples were taken every hour, representing a different
raw acid or metal concentration. Feed and water flow rates were set constant at 1.0
l/m²-hr. System and water temperatures were held constant at 30° and 40 °C
respectively. Recovery outlet level was constant at 41.5 cm.
## Evaluation of Diffusion Dialysis

The performance of the separation was evaluated according to procedures recommended by the Tokuyama Soda Co., LTD (1991). The parameters measured were: acid (by titration) and metal (by atomic absorption) concentration in the raw, recovered acid, and waste streams (Clesceri et al, 1989). Water, feed, recovered acid, and waste flow rates were also determined. The calculated parameters were: hydrostatic pressure difference across the membrane "ΔhP" (recovery hydrostatic pressure "hP<sub>rec</sub>" - waste hydrostatic pressure "hP<sub>was</sub>"), percentage of volume changes in recovered acid and waste streams, percentage of recovered acid, normality, and metal ion leakage.

## Feed Acid Concentration Variation

The concentration of sulfuric acid or nitric acid was increased using the system shown in Figure 2. The system allows the solution in the 200 ml stirred feed cylinder, to increase its acid concentration with time. The acid concentration in the cylinder can be estimated by the following equation

$$N_f = N_{stock} - N_{stock} e^{-Q_f / V_c t} \quad (1)$$

Where "N<sub>f</sub>" is the feed or raw normality, "N<sub>stock</sub>" is the normality of the stock solution, "Q<sub>f</sub>" is the raw flow rate, "V<sub>c</sub>" is the stirred cylinder volume, and "t" is the operating time. Therefore, the diffusion dialysis system could be run under a continuous increasing acid concentration. Recovered acid and waste samples were collected every 1 hr, representing different feed acid concentrations.

## Feed Metal Concentration Variation
The concentration of copper and iron were increased using the system shown in Figure 2. As mentioned before, the system is able to work under a continuous increasing feed metal concentration. The feed metal concentration can be estimated using equation (2).

$$F_{ppm} = STOCK_{ppm} - STOCK_{ppm} e^{-Q_f / V_J t} \quad (2)$$

Where "$F_{ppm}$" is the raw metal concentration, "$STOCK_{ppm}$" is the metal concentration in the stock solution. Samples were collected every 1 hr, to represent a different feed metal concentration performance.

# RESULTS AND DISCUSSION

## Feed Acid Concentration Variation

Figures 3 and 3A show the effect of increasing acid concentration in the feed over the hydrostatic pressure difference across the membrane "$\Delta hP$". The hydrostatic pressure difference increases exponentially as the raw acid concentration increases, shown as the area between the two lines ($hP_{rec}$ and $hP_{was}$), where "$hP_{rec}$" and "$hP_{was}$" is the recovery and waste side hydrostatic pressure respectively. As we can see in both figures, the "$hP_{rec}$" increases at a higher rate compared with the "$hP_{was}$". The difference can be calculated by comparing the derivatives from equation (5) and (6) for sulfuric acid and equation (7) and (8) for nitric acid. Sulfuric acid presents a "$\Delta(dP_{H2SO4}/dN) = 44.19$" and nitric acid a "$\Delta(dP_{HNO3}/dN) = 51.01$". The higher rate in $hP_{rec}$ is related to a higher recovery density rate. There are two different ways in
which density can be affected; anion (acid) transfer from the raw into the recovery,
and water transfer from the recovery into the feed side. From Figures 4 and 4A we
can see that the recovery volume decreases as the raw acid concentration increases.
Therefore, water is being transported from the recovery side into the waste side, a
process called osmosis. The increase in osmosis correlates well with the increase in
hydrostatic pressure difference. The sequence of events to achieve an increase in ΔhP
could be as follows; the hydrostatic pressure difference increases from an increase in
the recovery hydrostatic pressure, the recovery hydrostatic pressure increases from an
increase in density, the density increases from an increase in osmosis, and finally the
osmosis increases from an increase in solute concentration difference across the
membrane and an increase in ΔhP. These series of events have an impact over the
diffusion dialysis performance. Figures 5 and 5A show the effect of increasing raw
acid concentration over the recovered acid percentage, recovered acid normality, and
metal leakage percentage. The transport of anions (% Recovery) decreases linearly (r
= 0.98 sulfuric acid, and r = 0.96 nitric acid), cations (% Leakage) decrease
linearly for the copper in nitric acid (r = -0.94) and logarithmic for the iron in
sulfuric acid (r = -0.97), and the normality (% Normality) decreases linearly for
sulfuric acid (r = -0.90) and logarithmic for nitric acid (r = -0.98), with the increase
of raw acid concentration. The reduction in anion and cation transport across the
membrane can be explained by the extended form of the Nernst-Planck flux equation
$$J_i = C_i v^* - D_i \left[ \frac{dC_i}{dx} + C_i \frac{d \ln \gamma_i}{dx} + z_i \frac{F}{RT} C_i \frac{dE}{dx} + \frac{C_i}{RT} \left( V_i - \frac{M_i}{M_w} V_w \right) \frac{dP}{dx} \right] \quad (3)$$

where "$J_i$" is the ion "i" flux, "$C_i$" is the ion "i" concentration, "$v^*$" is the velocity of movement of the center of gravity of the pore liquid, "$D_i$" is the diffusivity of ion "i", "$x$" is the membrane thickness, "$\gamma_i$" is the activity coefficient for ion "i", "$z_i$" is the ion charge, "$F$" is the Faraday's constant, "$E$" is the electrical potential, "$V_i$ and $V_w$" are partial molar volumes for ion "i" and solvent "W" respectively, "$Mi$ and $MW$" are the ion and solvent molecular weight respectively, and "$P$" is the hydrostatic pressure. We can see that by increasing the hydrostatic pressure difference across the membrane, in an opposite direction of the flux, "$J_i$" will decrease, also if we increase the velocity "$v^*$" in an opposite direction of the flux, "$J_i$" will decrease as well.

These two terms appeared to be stronger than the rest of the terms in equation (3), since the overall anion flux decreases with an increase in raw acid concentration. In the case of nitric acid and copper, the hydrostatic pressure difference plays a more important role in the transfer of solute, since the osmosis levels off at very low raw acid concentrations. It is expected that adjusting the recovery outlet level to a lower level, should decrease the hydrostatic pressure difference, and therefore increase the acid flux. In the case of cation transport, the electrical repulsion by the charged membrane affects also the reduction in the cation flux, it is obvious by Figures 5 and 5A that there is more resistance to iron flux than copper flux. The fact that the normality percentage decreases with an increase in raw acid concentration, suggests
that anions are being transferred with the water flow (convective mass transfer) from the recovery side into the feed.

## Feed Metal Concentration Variation

### Sulfuric Acid / Ferrous Sulfate Solution

Figure 6 shows the effect of raw (sulfuric acid 10 N) iron concentration over the pressure difference across the membrane. The pressure difference across the membrane remains constant ($ΔhP_{H2SO4} = 516 ± 31 pa$) as the concentration of iron increases. Several factors, as we mentioned before, affect the hydrostatic pressure difference across the membrane. The main component is the density of the solutions, which in turn depends on the anion, cation, and water diffusion across the membrane. The cation concentration appears to have a small effect on the density. In the case of sulfuric acid and iron, solutions of 20,000 ppm of iron increase the density by 46.94 Kg/m³, which is equivalent to an increase of 19.09 pa in hydrostatic pressure.

Therefore, the hydrostatic pressure difference on either side of the membrane, is affected mainly by anion and water concentration and diffusion. As we can see from Figure 7, the recovery volume increases slightly with increasing iron concentration. The increasing recovery volume correlates with an increase in sulfate diffusion, as we can see from equation (17) ($dV_{rec}/dFe$) = 0.0002 and from equation (21) ($d%Rec/dFe$) = 0.00034. Therefore, we assume the increase in volume is related to the water of hydration from the diffusion of sulfate ions. It is known that the electrical field of a typical ion in solution, tends to attract a certain number of water molecules around the ion. This phenomena is called ion hydration, and it is of
interest for the reason that it is of considerable importance in ion diffusion as well as electroosmotic transfer (Tuwiner, 1962; Kolthoff and Elving, 1978). Our results show values of water of hydration for sulfate anions in the order of 0.5 moles of water per mol of sulfate. Maximum values of water of hydration under very diluted conditions are in the order of 5.5 moles of water per mol of sulfate (Horvath, 1985; Marcus, 1985). The lower value of water of hydration obtained in the sulfuric acid diffusion dialysis, must be related to the high concentration of acid we are dealing with. When anions are placed in concentrated solutions, the anions compete with each other for water molecules, this behavior reduces the water of hydration number, and therefore the recovered acid volume. Since the sulfuric acid recovery volume increases (Figure 7) and the sulfuric acid normality percentage is constant at % Normality = 76.77 ± 0.98 (Figure 8), with increases in raw iron concentration. The diffusion of sulfates (% Recovery) must increase as the raw iron concentration increases, to compensate for the dilution factor (Figure 8). This phenomena can be explained by the use of the Donnan equilibrium and the electroneutrality rule. The Donnan equilibrium for the diffusion dialysis of sulfuric acid and iron can be characterized as follows:

$$aSO_4 R = \left( \left[ \frac{aH R}{aH W} [aSO_4 W]^{0.5} \right]^2 \right) \quad (4)$$

where; "aSO₄ R" is the activity of sulfate in the recovery, "aSO₄ W" is the activity of the sulfate in the waste, "aH R" is the activity of the hydrogen in the recovery, and
"aH W" is the activity of the hydrogen in the waste. This equilibrium can be affected by the presence of a metal that is not allowed to cross the membrane. The concentration of hydrogen in the waste side is always related to the concentration of anions (sulfates) remaining, to maintain electroneutrality. As the concentration of the metal increases the hydrogen diffuses into the recovery side by difference in concentration, since iron will replace hydrogen to maintain electroneutrality.

Therefore, the hydrogen activity ratio increases, increasing the activity of sulfates on the recovery side in accord with Donnan equilibrium (Cwirko and Carbonell, 1990; Lake and Melshedaer, 1978; Wallace, 1967; Mears, 1986; Kessler and Klein, 1992).

Sato and coworkers (1985) have found the same effect in mixtures of nitric and hydrofluoric acid with increasing concentrations of iron. Figure 8 shows also the effect of raw iron concentration on the iron leakage percentage. The leakage percentage remains constant (% Leakage = 3.86) as the concentration of iron increases in the feed stream. The diffusion of iron across the membrane can be explained by two theories; 1.) Iron can cross the charged membrane when it is bound to a sulfate ion, and therefore the charge of the molecule becomes neutral. Salts with low equilibrium constants (weak salts), have a higher concentration of the metal in a neutral state, allowing higher leaking across the membrane. Ferrous sulfate has a high equilibrium constant

$$FeSO_4 \rightarrow SO_4^{++} + Fe^{=} \quad K_{Fe} = 0.04 \quad (Bjerrum et al, 1958)$$

which correlates with a low percentage of leakage. 2.) Iron may form hydrates with water, which also present a neutral charge. But, its volume becomes so high that
stearic limitations prevent the molecules from crossing the membrane. Iron presents a hydrated volume "V<sub>fe</sub>" of 174.5 - 198 cm³/mol, with a radius "r<sub>Fe</sub>" of 4.28 Å (Horvath, 1985; Marcus, 1985). Both characteristics correlate well with the low percentage of leakage observed.

## Nitric Acid / Copper Nitrate Solution

Figure 6A shows the effect of raw copper concentration over the hydrostatic pressure across the membrane in diffusion dialysis of nitric acid. The hydrostatic pressure difference decreases as the copper concentration increases, mainly from an increase in the waste hydrostatic pressure. The waste hydrostatic pressure increases linearly (r = 0.90), and the recovery hydrostatic pressure remains constant (hP<sub>rec</sub> = 4780.66 ± 7.35 pa), as the raw copper concentration increases. As we mentioned before, the hydrostatic pressure depends on the anion and water concentration and diffusion. Since the density of the waste increases with feed stream copper concentration (higher hP<sub>wash</sub>), there must be a decrease in anion diffusion (from the feed into the recovered acid) or a decrease in osmosis (from the recovered acid into the feed). Since Figure 6A shows that the recovery volume decreases with increasing feed copper concentration, the metal concentration must be contributing with an increase in osmotic pressure. The increase in "hP<sub>wash</sub>" must be from a decrease in anion diffusion. Since osmosis increases with feed stream copper concentration, the anion diffusion must decrease in order to maintain an increasing hydrostatic pressure on the waste side. Figure 8A shows that the recovery percentage for nitric acid decreases with increasing raw copper concentration. The percentage of nitric acid
normality is maintained constant, although diffusion of nitric acid decreases due to an increase in osmosis. Nitric acid and copper present an opposite behavior patterns when compared with sulfuric acid and iron. Donnan equilibrium does not have a great deal of influence in the nitric acid diffusion for two reasons; 1.) Copper is not isolated from the recovery side as well as iron. The leakage in the case of copper goes as high as 35.5 %. Therefore, the hydrogen activity ratio does not increase greatly. 2.) From equation 4 we can see that the hydrogen activity ratio is elevated to a single power for sulfuric acid, while in the case of nitric acid the hydrogen ratio would be elevated to a second degree, this also reduces the effect over the nitrates activity in the recovery side. In addition, the nitric acid recovery is also affected by the copper leakage since it decreases the activity of hydrogen on the recovery stream. Copper replaces hydrogen to maintain electroneutrality in the recovery stream, therefore an increasing amount of nitrates is in a copper nitrate state, as copper concentration increases in the feed stream. The higher leakage of copper in nitrate solution, in comparison to iron in sulfate solution, can be related to the lower dissociation constant ($K_{a} = -16$) and a lower hydrated volume ($h_{vA} = 185 cm^3/mol$, $h_{vF} = 198 cm^3/mol$) (Bjerrum et al, 1958; Horvath, 1985; Marcus, 1985). It is our feeling, that the equilibrium constant is a more important factor than hydrated volume, when it comes to metal leakage.
CONCLUSION

The concentration of acid in the feed stream plays an important role in the efficiency of the acid diffusion dialysis separation. When the recovery outlet level is maintained constant, the hydrostatic pressure difference across the membrane increases, reducing the recovered acid volume and percentage of recovered acid. The main reason for this phenomena is the increase in the recovery density with increasing acid concentration in the feed stream. The reduction in acid flux can be explained by the use of the extended form of the Nernst-Planck flux equation. Iron concentration in the sulfuric acid feed appears to have a small effect on the hydrostatic pressure difference, mainly because the density changes are negligible. But, iron has an important effect on sulfate ion diffusion. Iron in sulfuric acid has a very high salt equilibrium constant and ion hydrated volume. These characteristics provide separation with very low iron leakage. Having a good selectivity against the iron, the system depends upon Donnan equilibrium. The increasing iron concentration in the feed stream, increases the hydrogen activity ratio in the Donnan equilibrium, which in turn increases the sulfate activity in the recovery (higher sulfate flux). Nitric acid and copper nitrate present an opposite behavior compare with sulfuric acid and iron sulfate. The main difference between them is the higher metal leakage by the copper in nitric acid. The higher leakage can be related to a low salt equilibrium constant and low copper ion hydrated volume. A high copper leakage and a monovalent relationship between hydrogen and nitrate, make a weak Donnan equilibrium. Therefore the nitrate flux does not increase with higher copper concentration.
REFERENCES

Bailey, D., and Howard, T., Acid Recovery with Diffusion Dialysis, *Metal Finishing*, 1992, November, 21-23.

Bailey, D.E., Recovery and Reuse of Spent Acid Baths via Diffusion Dialysis, *Hazardous and Municipal Waste Minimization Conference, Providence RI Feb.*
7-8, 1989.

Bejerrum, J., Schwarzenbach, G., and Sillen, L.G., Stability Constants of Metal-Ion Complexes, Special Publication No.7, *The Chemical Society, Burlington House, London 1957, 29-43.*

Clesceri, S.L., Greenberg, E.A. and Trussell, R.R., *Standard Methods, For the Examination of Water and Wastewater*, American Public Health Association, Washington DC 1989, 2-31,3-13.

Cwirko, E.H. and Carbonell, R.G., A Theoretical Analysis of Donnan Dialysis Across Charged Porous Membranes, *Journal of Membrane Science*, 1990, 48, 155-179.

Fritsch, J., Scharff, K., Schmid, G., and Strathmann, H., Elektrodialyse und Diffusionsdialyse, *Galvanotechnik*, 1991, 82, 7, 2298-2307.

Hikari, Kashimacho, Kashima-gun, and Ibaraki, Pickling Acid-recovery Process by Diffusion Dialysis, *New Technology*, 1987, 386.

Horvath, A.L., *Handbook of Aqueous Electrolyte Solutions*, John Wiley and Sons, NY 1985, 344.

Kessler, S.B. and Klein, E., *Membrane Handbook*, ed. W.S. Winston Ho and K.K. Sirkar, Van Nostrand Reinhold, NY 1992, 161-215.

Klein, E., Ward, R.A. and Lacey, R.E., *Handbook of Separation Process Technology*, ed. R.W. Rousseau, John Wiley & Sons, NY 1987, 954-981.

Kolthoff, M.I., and Elving, P.J., *Treatise on Analytical Chemistry, Part 1 Theory and Practice, 2nd ed.*, John Wiley and Sons, NY 1978, 556.

Lake, A. M., and Melshedaer, S.S., Mass Transfer Characterization of Donnan Dialysis, *American Institute of Chemical Engineers Journal*, 1978, 24, 1, 130-137.
Lakshminarayanaiah, N., Transport Phenomena in Membranes, Academic Press, NY & London 1969, 58-361.

Linch, J.D., Acid Recovery Via Dialysis, HPD Incorporated, Membrane Technology/Planning Conference, 1985.

Masayuki, O., Uchida, M., and Ishibashi, T., Stainless Steel Pickling by Fixed Concentration HF/HNO₃ Mixtures Through Diffusion Dialysis Acid Reclamation, Wire Journal International, 1992, March, 43-48.

Marcus, Y., Ion Solvation, John Wiley and Sons, NY 1985, 100-101.

Mears, P., Synthetic Membranes: Science, Engineering and Applications, ed. P.M. Bungay, H.K. Lonsdale and M.N. de Pinho, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo 1986, 169-179.

Nakamura, I., Kawahara, T., and Utsunomiya, T., Use of Dialysis Techniques in Metal Finishing Process, New Materials and New Processes, 1983, 2, 166-175.

Sato, J., Onuma, M., Motomura, H. and Noma, Y., Recovery of Nitric Acid and Hydrofluoric Acid from the Pickling Solution by Diffusion Dialysis, Tokuyama Soda Co., LTD, Reprint from Jitsumu Hyomen Gijutsu, The Metal Finishing Society of Japan, 1985, 1-8.

Sridhar, P., and Subramaniam, G., Recovery of Acid From Cation Exchange Resin Regeneration Waste by Diffusion Dialysis, Journal of Membrane Science, 1989, 45, 273-280.

Tokuyama Soda CO., LTD., Instruction Manual for Diffusion Dialysis Unit Model TSD-2, Tokuyama Japan 1991.

Tuwiner, S.B., Diffusion and Membrane Technology, Reinhold Publishing Corporation, NY & London 1962, 38-60, 77-90, 137-151, 287-320.

Wallace, R.M., Concentration and Separation of Ions by Donnan Membrane Equilibrium, Industrial and Engineering Chemistry, Process Design and Development, 1967, 6, 423-431.
<figure><img src="image_30.png" /></figure>
<figure><img src="image_31.png" /></figure>
<figure><img src="image_32.png" /><figcaption>Figure 3
Effect of sulfuric acid concentration on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$r = 0.99$

$$P_{\text{rec}} = 77.58N + 4067.55 \quad (5)$$

Waste Pressure Linear Regression

$r = 0.98$

$$P_{\text{was}} = 33.39N + 4029.83 \quad (6)$$
<figure><img src="image_33.png" /><figcaption>Figure 3A
Effect of nitric acid concentration on pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$$r = 0.99$$
$$P_{\text{rec}} = 78.45N + 4090.02 \quad (7)$$

Waste Pressure Linear Regression

$$r = 0.98$$
$$P_{\text{wax}} = 27.44N + 4035.75 \quad (8)$$
<figure><img src="image_34.png" /><figcaption>Figure 4
Effect of sulfuric acid on
recovery and waste volumes</figcaption></figure>

Statistical Analysis

Recovery Volume Linear Regression

$$
\begin{gathered}
r = 0.84 \\
e^{\text{VolRec}} = -4.3996 \times 21 \, N + 3.2987 \times 22 \quad (8a)
\end{gathered}
$$

Waste Volume Linear Regression

$$
\begin{gathered}
r = 0.97 \\
\ln \text{Vol}_{\text{was}} = 0.0518N + 3.7729 \qquad (8b)
\end{gathered}
$$
<figure><img src="image_35.png" /><figcaption>Figure 4A
Effect of nitric acid on
recovery and waste volumes</figcaption></figure>

Statistical Analysis

Recovery Volume 2nd Order Regression

$r = 0.93$

$$V_{Rec} = 0.13N^2 - 1.27N + 37.80 \quad (8c)$$

Waste Volume 2nd Order Regression

$r = 0.93$

$$V_{Was} = -0.13N^2 + 1.27N + 62.20 \quad (8d)$$
<figure><img src="image_36.png" /><figcaption>Figure 5
Effect of sulfuric acid concentration on
acid recovery, normality and Cu leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Linear Regression

$$
\begin{align*}
r &= 0.98 \\
\% \text{ Rec} &= -5.46N + 100.80 \tag{9}
\end{align*}
$$

Normality Percentage Linear Regression

$$
\begin{align*}
r &= 0.90 \\
\ln \%N &= -1.38N + 92.28 \tag{10}
\end{align*}
$$

Leakage Percentage Linear Regression

$$
\begin{align*}
r &= 0.97 \\
\ln \text{\% Leakage} &= -0.3597N + 4.021 \tag{11}
\end{align*}
$$
<figure><img src="image_37.png" /><figcaption>Figure 5A
Effect of nitric acid concentration on
acid recovery, normality and Cu leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Linear Regression

$$
\begin{gathered}
r = 0.96 \\
\% \text{ Rec} = -2.61N + 87.64 \quad (12)
\end{gathered}
$$

Normality Percentage Linear Regression

$$
\begin{gathered}
r = 0.98 \\
\ln \%N = -0.037N + 4.83 \quad (13)
\end{gathered}
$$

Leakage Percentage Linear Regression

$$
\begin{gathered}
r = 0.94 \\
\% \text{ Leakage} = -0.927N + 22.088 \quad (14)
\end{gathered}
$$
<figure><img src="image_38.png" /><figcaption>Figure 6
Effect of iron concentration on the pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Average and Standard Deviation
$P_{rec} = 4763.28 \pm 8.84$

Waste Pressure Average and Standard Deviation
$P_{was} = 4246.85 \pm 23.22$

Pressure Difference Average and Standard Deviation
$\Delta P = 516.42 \pm 31.12$
<figure><img src="image_39.png" /><figcaption>Figure 6A
Effect of copper concentration on the pressure difference across the membrane</figcaption></figure>

Statistical Analysis:

Recovery Pressure Average and Standard Deviation

$$P_{\text{rec}} = 4780.66 \pm 0.23$$

Waste Pressure Linear Regression

$r = 0.90$

$$P_{\text{was}} = 0.00083N + 4123.28 \quad (15)$$

Pressure Difference Linear Regression

$r = 0.73$

$$\Delta P = -.00072N + 654.52 \quad (16)$$
<figure><img src="image_40.png" /><figcaption>Figure 7
Effect of iron concentration on
recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Linear Regression

$r = 0.92$

$$V_{\text{rec}} = 0.00020N + 50.82 \quad (17)$$

Waste Volume Linear Regression

$r = 0.92$

$$V_{\text{was}} = -0.00020N + 49.17 \quad (18)$$
<figure><img src="image_41.png" /><figcaption>Figure 7A
Effect of copper concentration on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Linear Regression

$$r = 0.83$$
$$V_{rec} = -0.000075N + 59.01 \quad (19)$$

Waste Volume Linear Regression

$$r = 0.83$$
$$V_{wast} = 0.000075N + 40.98 \quad (20)$$
<figure><img src="image_42.png" /><figcaption>Figure 8
Effect of iron concentration on acid recovery, normality and iron leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Linear Regression

$$r = 0.94$$
$$\% \text{Rec} = 0.00034N + 76.87 \quad (21)$$

Normality Percentage Average and Standard Deviation

$$\% N = 76.77 \pm 0.98$$

Iron Leakage Percentage Average and Standard Deviation

$$\% \text{Leakage} = 3.86 \pm 0.49$$
<figure><img src="image_43.png" /><figcaption>Figure 8A
Effect of copper concentration on acid recovery, normality and copper leakage</figcaption></figure>

Statistical Analysis:

Recovery Percentage Linear Regression

$r = 0.92$

$$ \% \text{Rec} = -0.000095N + 94.11 \quad (22) $$

Normality Percentage Linear Regression

$$ \% N = 81.63 \pm 0.89 $$
# Chapter V
HYDROSTATIC PRESSURE DIFFERENCE EFFECT ON
DIFFUSION DIALYSIS PERFORMANCE
SUMMARY

Diffusion dialysis is a membrane technology process that provides an alternative to neutralization, traditional treatment of spent acid baths. Diffusion dialysis performance is evaluated based on the acid recovery, acid purity, and acid concentration. These parameters are affected by the hydrostatic pressure difference across the membrane. It is our objective to provide an understanding of the effect of the hydrostatic pressure difference on the performance of diffusion dialyzer, as well as a way to control it. Sulfuric acid / ferrous sulfate and nitric acid / copper nitrate solutions were treated with an anion diffusion dialysis system. The effect of recovery outlet level on percentage of: recovered acid, recovered normality, and metal leakage were studied. Increasing the recovery outlet level increases the hydrostatic pressure difference across the membrane. The increase in hydrostatic pressure difference across the membrane decreases the recovered acid volume, which increases the normality and decreases the metal leakage and acid recovery. The convective water flow in an opposite direction to the acid diffusion may be the factor responsible for reducing the anion flux across the membrane. This theory is in accordance with the extended form of the Nernst-Planck flux equation. The ability to adjust the recovery outlet level was found to be a great way of controlling dilution and high hydrostatic pressure problems. The optimum recovery outlet level will depend on the specific acid solution and process, a compromise between concentration, recovery and metal leakage is inevitable
INTRODUCTION

Diffusion dialysis (DD) is a membrane technology process that provides an
alternative to the traditional method (neutralization) for treatment of spent acid baths
(Masayuki, 1992; Nakamura, 1983). DD is able to separate acids (anions) from
metals (cations) by the use of an anion-exchange membrane, as illustrated in Figure 1.
The acid can be recycled back into the process (Bailey and Howard, 1992; Hikari et
al, 1987; Linch, 1985; Fritsch et al, 1991). Three major advantages of returning the
acid are; 1.) less acid consumption, 2.) less alkaline consumption, and 3.) less sludge
production. These advantages provide a good incentive for industry to introduce DD
into their process. DD performance can be evaluated based on the following
parameters; amount of recovered acid, purity of the recovered acid, and concentration
of recovered acid (Sato et al, 1985; Sridhar and Subramaniam, 1989). The ideal
performance is one that provides the highest recovery, concentration, and purity. In
reality these parameters counter each other, and we can not get the highest separation
with the highest concentration and the highest purity. Therefore, we must understand
how these parameters change with different conditions if we desire to optimize the
separation. Some of the variables that are known to affect the performance of DD
are; temperature, acid and metal concentration, flow rate, and hydrostatic pressure
(Bailey, 1989). The latest is the focus of our paper, and it is our intention to provide
an understanding of the effect of hydrostatic pressure in DD and a way to control it.
The hydrostatic pressure in a DD compartment will depend on the density of the
solution, and the length of the liquid column. DD systems are fabricated in two
geometric designs: horizontal and vertical. In the horizontal design, the membrane stack is arranged so that the stream compartments are perpendicular to the gravitational force. In the vertical design, the membrane stack is arranged so that the stream compartments are parallel to the gravitational force. The latest design is the must popular one and is shown in Figure 1. Fluid mechanics in the vertical design is such that a recovery liquid column is needed in order to maintain fluid level of the recovery compartment. This column, shown in Figure 1 as "REC", provides the recovery compartment with a hydrostatic pressure that depends on the density and length of the column. At the other side of the membrane (feed side), the hydrostatic pressure will also depend on the density and length of the feed/waste column. The hydrostatic pressure difference across the membrane, will be the difference between the recovery and feed hydrostatic pressure. The relevance of the hydrostatic pressure may depend on the water permeability of the membrane, which in turn is known to be related to the membrane water content. High water content in the membrane correlates well with high water permeability, since diffusion occurs only through the water passases. Therefore, the effect of the hydrostatic pressure in DD may be crucial for high water content membranes but not so important to low water content ones. When an excess hydrostatic pressure is applied to the solution on one side, the equilibrium in the pores of the membrane is displaced leading to the development of a convective flow, in the direction of reducing pressure. Schlögl (1966) has given a completely extended form of the Nernst-Planck flux equation including the pressure term and the convection term, equation (1) (Lakashminarayanaiah, 1969).
$$J_i = C_i v^* - D_i \left[ \frac{dC_i}{dx} + C_i \frac{d \ln \gamma_i}{dx} + z_i \frac{F}{RT} C_i \frac{dE}{dx} + \frac{C_i}{RT} \left( V_i - \frac{M_i}{M_w} V_w \right) \frac{dP}{dx} \right] \quad (1)$$

The simultaneous presence of both concentration and pressure driving forces results in a mixed diffusive-convective transport process. When the two fluxes are significant, both must be accounted for, as well as their interactions. Another way of analyzing mass transfer under these conditions is irreversible thermodynamics. The hydraulic permeability of a membrane "Lₚ", is expressed in the form of a coefficient of proportionality between volumetric flux "Jₜ" and pressure difference across the membrane "ΔP" as shown in equation (2)

$$J_v = L_p \Delta P \quad (2)$$

In the presence of osmotic pressure difference, equation (2) becomes:

$$J_v = L_p \Delta P + L_{pD} \Delta \pi \quad (3)$$

where "LₚD" is the coefficient of osmotic flow, and "ΔΠ" is the osmotic pressure difference across the membrane (Kessler and Klein, 1992; Lakashminarayanaiah, 1969). Equation (3) shows why the water permeability of the membrane is so important when it comes to changes in hydrostatic pressure and osmotic pressure.
# EXPERIMENTAL

## Acid Solutions

Nitric and sulfuric acid solutions were prepared according to the following acid and metal concentrations:

1) Sulfuric acid 10 N / Iron 11,000 ppm

2) Nitric acid 10 N / Copper 30,000 ppm

Acid solutions were prepared using analytical reagents from Fisher Scientific.

## Diffusion Dialysis Set Up

Diffusion dialysis separations were performed using a Tokuyama Soda Co., LTD., model TSD5-1, with an anion-exchange membrane Neosepta model AFN. Figure 2 shows the experimental set up used for DD runs. All of the runs were repeated at least twice. The recovery outlet level is measured from the feed inlet to the recovery outlet, shown as Hrec in Figure 1. The recovery outlet level was adjusted to different lengths, to provide an adequate recovery hydrostatic pressure. Samples were collected for 1 hr after a minimum running time of 5 h. Feed and water flow rates were set constant at 1.0 l/m²-hr. System and water temperatures were held constant at 30 and 40 °C respectively.

## Evaluation of Diffusion Dialysis

The performance of the separation was evaluated according to the procedures recommended by Tokuyama Soda Co., LTD (1991). The parameters measured were: acid (by titration) and metal (by atomic absorption) concentration in the feed, recovered acid, and waste streams Clesceri et al, 1989). Water, feed, recovered acid,
and waste flow rates were also determined by measuring the respective volumes after one hour of running. The calculated parameters were: hydrostatic pressure difference across the membrane "ΔhP" (recovery hydrostatic pressure "hP<sub>rec</sub>" - waste hydrostatic pressure "hP<sub>was</sub>"), percentage of volume changes in recovered acid and waste streams, percentage of recovered acid, normality, and metal ion leakage.

## RESULTS AND DISCUSSION

Figure 3 and 3A show the effect of recovery outlet level on the hydrostatic pressure difference across the membrane, for sulfuric/iron and nitric/copper respectively. The hydrostatic pressure difference "ΔhP = hP<sub>rec</sub> - hP<sub>was</sub>" increases and then levels off, by increasing the recovery outlet level. This phenomena is related to a higher initial slope by the hP<sub>rec</sub> and an exponential shape by the hP<sub>was</sub>. As we can see from Figures 3 and 3A the recovery hydrostatic pressure "hP<sub>rec</sub>" increases linearly (r = 0.99 for sulfuric acid, r = 0.99 for nitric acid) as the recovery outlet level increases. The waste hydrostatic pressure "hP<sub>was</sub>" increases exponentially (r = 0.98 for sulfuric acid/iron, (r = 0.92) for nitric acid/copper) as the recovery outlet level increases. The increase in hP<sub>rec</sub> with increasing recovery outlet level comes from an increase in the column length. The equation for the estimation of the hydrostatic pressure is

$$hP_{rec} = \rho gH \quad (4)$$

where "ρ" is the density of the fluid, g = 9.81 m/sec², and "H" is the column length,
in this case the recovery outlet level. In the case of the waste compartment, since the waste column length is fixed at all times the $hP_{\text{was}}$ increase must come from an increase in the density of the waste $"\rho_{\text{was}}"$. The change in the density of the waste compartment can come from a decrease in acid diffusion into the recovery side and or an increase of water diffusion into the recovery side. Volume measurements show that the water is actually diffusing into the waste side, therefore the increase in the waste density must come from a decrease in the acid diffusion. Figures 4 and 4A show the effect of recovery outlet level on the percentage of recovery and waste volume, for sulfuric/iron and nitric/copper respectively. Diffusion of water into the waste side increases exponentially ($r = 0.95$ for sulfuric/iron and $r = 0.83$ for nitric/copper) as the hydrostatic pressure difference increases. This phenomena has been already defined as convective flow (equation 3), by a pressure difference across a membrane (Lakashminarayanaiah, 1969; Kessler and Klein, 1992). The pressure difference can be hydrostatic and/or osmotic as equation (3) shows. In our case, the effect of osmotic pressure can be considered negligible compare to the hydrostatic pressure difference effect, since the recovery volume is greater than the waste volume at $\Delta hP = 0$ (Figure 4 and 4A). The higher volume in the recovery at low $\Delta hP$ could be related to the water carried by the anions when they cross the membrane (ion hydration) (Tuwiner, 1962; Kolthoff and Elving, 1978). Ion hydration appears to be stronger than osmotic pressure, and hydrostatic pressure difference appears to be stronger than both of them. Figures 5 and 5A show the effect of hydrostatic pressure difference on acid recovery, normality, and metal leakage. The percentage of
normality increases linearly ($r$ = 0.93 for sulfuric/iron and $r$ = 0.99 for nitric/copper), and the metal leakage decreases linearly ($r$ = 0.97 for sulfuric/iron and $r$ = 0.99 for nitric/copper), as the hydrostatic pressure difference increases. The normality of the recovered acid increases with increasing hydrostatic pressure difference and correlates well with the decrease in recovery volume. Decreasing the volume in the recovery should increase the acid concentration, assuming there is minimum acid flux along with the water flow going back into the waste side. The reducing metal leakage with increasing hydrostatic pressure difference, may be related to the increasing water flow in the opposite direction (recovery into the waste). The water flow adds a convective factor (convective term from equation 1) that reduces the metal flux. The same situation happens with the recovered acid. Acid recovery decreases logarithmically ($r$ = 0.84 for sulfuric/iron and $r$ = 0.89 for nitric/copper) as the hydrostatic pressure increases, which correlates very well with the logarithmic reduction in recovery volume. Again the reduction in the acid flux may be related to the convective flow going in an opposite direction. The extended form of the Nerst-Planck flux equation (1), shows the hydrostatic pressure and convective flow terms.

In equation (1)

$$dP/dx = \Delta hP = h_{rec} - h_{was} \quad (5)$$

$$C_i V' = C_{was} (Q_{was} - Q_{feed}) \quad (6)$$

where "$Q_{was}$" and "$Q_{feed}$" are the waste rate and feed flow rates respectively, assuming

$$dC = C_{rec} - C_{was} \quad (7)$$
therefore, as the hydrostatic pressure difference and the convective flow increases, the
anion flux "$J_i$" decreases.

CONCLUSION

On a vertical diffusion dialysis setup, the hydrostatic pressure difference across the membrane can be adjusted by modifying the recovery outlet level. When the recovery outlet level is increased the hydrostatic pressure difference increases, mainly from a higher increase in the recovery hydrostatic pressure than the waste hydrostatic pressure. High hydrostatic pressure differences across the membrane enhance a convective flow in the direction of reducing pressure. The convective flow decreases the recovery volume, which increases the recovered acid normality and decreases the percentage of recovered acid and metal leakage. The results are in accordance with the extended form of the Nernst-Planck flux equation, which considers the convective and hydrostatic pressure terms for the calculation of the ion flux.

There is a compromise between the recovered acid concentration and
the percentage of recovered acid and metal leakage. The capacity to adjust the
recovery outlet level to different hydrostatic pressures, increases the ability of the unit
to reach optimum conditions for the particular separation. There is no optimum
recovery outlet level for any acid solution. Every acid separation will have a
different set of conditions that affect the hydrostatic pressure difference; every acid
process will have different requirements in terms of concentration, recovery and
purity. Nevertheless the capacity to adjust the recovery outlet level is a tool that will
help any acid separation for whatever process.
REFERENCES

Bailey, D.E., Recovery and Reuse of Spent Acid Baths via Diffusion Dialysis, *Hazardous and Municipal Waste Minimization Conference, Providence RI Feb.*
7-8, 1989.

Bailey, D., and Howard, T., Acid Recovery with Diffusion Dialysis, *Metal Finishing*, 1992, November, 21-23.

Clesceri, S.L., Greenberg, E.A. and Trussell, R.R., *Standard Methods. For the Examination of Water and Wastewater*, Amercian Public Health Association, Washington DC 1989, 2-31,3-13.

Fritsch, J., Scharff, K., Schmid, G., and Strathmann, H., Electrodialyse und Diffusiondialyse, *Galvanotechnik*, 1991, 82, 7, 2298-2307.

Hikari, Kashimacho, Kashima-gun, and Ibaraki, Pickling Acid-recovery Process by Diffusion Dialysis, *New Technology*, 1987, 386.

Kessler, S.B. and Klein, E., *Membrane Handbook*, ed. W.S. Winston Ho and K.K. Sirkar, Van Nostrand Reinhold, NY 1992, 161-215.

Kolthoff, M.I., and Elving, P.J., *Treatise on Analytical Chemistry, Part I* Theory and Practice, 2nd ed., John Wiley and Sons, NY 1978, 556.

Lakshminarayanaiah, N., *Transport Phenomena in Membranes*, Academic Press, NY & London 1969, 58-361.

Linch, J.D., Acid Recovery Via Dialysis, HPD Incorporated, *Membrane Technology/Planning Conference*, 1985.

Masayuki, O., Uchida, M., and Ishibashi, T., Stainless Steel Pickling by Fixed Concentration HF/HNO₃ Mixtures Through Diffusion Dialysis Acid Reclamation, *Wire Journal International*, 1992, March, 43-48.

Nakamura, I., Kawahara, T., and Utdunomiya, T., Use of Dialysis Techniques in Metal Finishing Process, *New Materials and New Processes*, 1983, 2, 166-175.

Sato, J., Onuma, M., Motomura, H. and Noma, Y., Recovery of Nitric Acid and Hydrofluoric Acid from the Pickling Solution by Diffusion Dialysis, Tokuyama Soda Co., LTD, Reprint from Jitsumu Hyomen Gijutsu, *The Metal Finishing Society of Japan*, 1985, 1-8.
Schlögl, R., *Berichte der Bunsengesellschaft fuer Physikalische Chemie*, 1966, 71, 755.

Sridhar, P., and Subramaniam, G., Recovery of Acid From Cation Exchange Resin Regeneration Waste by Diffusion Dialysis, *Journal of Membrane Science*, 1989, 45, 273-280.

Tokuyama Soda CO., LTD., *Instruction Manual for Diffusion Dialysis Unit Model TSD-2*, Tokuyama Japan 1991.

Tuwiner, S.B., *Diffusion and Membrane Technology*, Reinhold Publishing Corporation, NY & London 1962, 38-60, 77-90, 137-151, 287-320.
<figure><img src="image_44.png" /></figure>
<figure><img src="image_45.png" /></figure>
<figure><img src="image_46.png" /><figcaption>Figure 3
Sulfuric acid 10 N, recovery outlet
level effect on hydrostatic pressure</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$$
\begin{align*}
r &= 0.99 \\
P_{\text{rec}} &= 143.34l - 1137.36 \tag{8}
\end{align*}
$$

Waste Pressure Linear Regression

$$
\begin{align*}
r &= 0.98 \\
P_{\text{WAS}} &= 72.28l + 1303.51 \tag{9}
\end{align*}
$$
<figure><img src="image_47.png" /><figcaption>Figure 3A
Nitric acid 10 N, recovery outlet level
effect on hydrostatic pressure</figcaption></figure>

Statistical Analysis:

Recovery Pressure Linear Regression

$$
\begin{gathered}
r = 0.99 \\
P_{\text{rec}} = 132.29l - 681.02 \qquad (10)
\end{gathered}
$$

Waste Pressure Linear Regression

$$
\begin{gathered}
r = 0.92 \\
\ln P_{\text{was}} = 0.0111l + 7.9046 \qquad (11)
\end{gathered}
$$
<figure><img src="image_48.png" /><figcaption>Figure 4
Sulfuric acid 10 N, membrane delta "P" effect on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Percentage Linear Regression

$r = 0.82$

$$e^{Vol \ Rec} = 6.1980 E 29 l + 6.1980 E 29 \quad (12)$$

Waste Volume Percentage Linear Regression

$r = 0.95$

$$\ln Vol_{Was} = 0.0028l + 2.67 \quad (13)$$
<figure><img src="image_49.png" /><figcaption>Figure 4A
Nitric acid 10 N, membrane delta "P" effect on recovery and waste volumes</figcaption></figure>

Statistical Analysis:

Recovery Volume Percentage Linear Regression

$r = 0.69$

$$e^{Vol \ Rec} = -4.0737 \ E^{23.1} + 1.4703 \ E^{26} \quad (14)$$

Waste Volume Percentage Linear Regression

$r = 0.83$

$$\ln Vol_{Was} = 0.0011l + 3.5671 \quad (15)$$
<figure><img src="image_50.png" /><figcaption>Figure 5</figcaption></figure>

Sulfuric acid 10 N, membrane delta "P" effect on
acid recovery, normality, and iron leakage

Statistical Analysis:

Recovery Percentage Linear Regression

$$
\begin{gathered}
r = 0.89 \\
e^{\%Rec} = -4.0525 \ E \ 39 \ l \ + \ 1.7598 \ E \ 42
\end{gathered}
\qquad (16)
$$

Normality Percentage Linear Regression

$$
\begin{align*}
r &= 0.99 \\
\%N &= 0.044l + 58.02 \tag{17}
\end{align*}
$$

Leakage Percentage Linear Regression

$$
\begin{align*}
r &= 0.99 \\
\text{Leakage \%} &= -0.1873l + 92.4628 \tag{18}
\end{align*}
$$
<figure><img src="image_51.png" /><figcaption>Figure 5A</figcaption></figure>

· Nitric acid 10 N, membrane delta "P" effect on

· acid recovery, normality, and copper leakage

Statistical Analysis:

Recovery Percentage Linear Regression

$$
\begin{gathered}
r = 0.89 \\
e^{\%Rec} = -4.0525l + 1.7598 E 42 \quad (19)
\end{gathered}
$$

Normality Percentage Linear Regression

$$
\begin{align*}
r &= 0.99 \\
\%N &= 0.044l + 58.02 \tag{20}
\end{align*}
$$

Leakage Percentage Linear Regression

$$
\begin{gather*}
r = 0.99 \\
\text{Leakage \%} = -0.1873l + 92.4628 \tag{21}
\end{gather*}
$$
# CHAPTER VI
BIBLIOGRAPHY

Atkins, P.W., *Physical Chemistry*, Fourth ed., W.H. Freeman and Company, NY
1990, 737, 761-769, 792.

Bailey, D.E., Recovery and Reuse of Spent Acid Baths via Diffusion Dialysis,
*Hazardous and Municipal Waste Minimization Conference*, Providence RI Feb.
7-8, 1989.

Bailey, D., and Howard, T., Acid Recovery with Diffusion Dialysis, *Metal
Finishing*, 1992, November, 21-23.

Bird, R.B., Stewart, W.E., and Lightfoot, N.E., Transport Phenomena, Jhon Wiley
and Sons, NY 1960, 504-505.

Bjerrum, G.S., and Sillen, L.G., Stability Constants of Metal-Ion Complexes, with
Solubility Products of Inorganic Substances, Part II: Inorganic Ligands, The
*Chemical Society*, Burlington House 1958, Special Publication No.7, 55,43,82-
83.

Bungay, P.M., Lonsdale, H.K., and de Pinho, M.N., Synthetic Membranes:
*Science, Engineering and Applications*, ed. D. Reidel Publishing Company,
Dordrecht 1986, 109-112.

Clesceri, S.L., Greenberg, E.A. and Trussell, R.R., Standard Methods, For the
*Examination of Water and Wastewater*, American Public Health Association,
Washington DC 1989, 2-31,3-13.

Cwirko, E.H. and Carbonell, R.G., A Theoretical Analysis of Donnan Dialysis
Across Charged Porous Membranes, *Journal of Membrane Science*, 1990, 48,
155-179.

Davis, T.A., Application of Ion-Exchange Membranes to Industrial Separations,
*Consultant Information*, NJ 1992, 1-7.

Denbigh, K., *The Principles of Chemical Equilibrium*, 4th ed., Cambridge
University Press, London 1982, 116-119.

Drukker, W., Parsons, F.M., and Maher, J.F., Replacement of Renal Function by
*Dialysis*, Second ed., Martinus Nijhoff, Boston MA 1983.

Fritsch, J., Scharff, K., Schimid, G., and Strathmann, H., Elektrodialyse und
Diffusionsdialyse, *Galvanotechnik*, 1991, 82, 7, 2298-2307.
Graver Water, Division of The Graver Company, **Membrane Technologies**, Diffusion Dialysis, Union NJ 1993.

Hikari, Kashimacho, Kashima-gun, and Ibaraki, Pickling Acid-recovery Process by Diffusion Dialysis, **New Technology**, 1978, 386.

Horvath, A.L., **Handbook of Aqueous Electrolyte Solutions**, Jhon Wiley and Sons, NY 1985, 344.

Johnson, K.R., Annalen der Physik, Leipzig 1904, 14, 995.

Jonsson, G., Synthetic Membranes: Science, Engineering and Applications, ed. P.M. Bungay, H.K. Lonsdale and M.N. de Pinho, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo 1986, 625-645.

Kessler, S.B. and Klein, E., **Membrane Handbook**, ed. W.S. Winston Ho and K.K. Sirkar, Van Nostrand Reinhold, NY 1992, 161-215.

Klein, E., Ward, R.A. and Lacey, R.E., **Handbook of Separation Process Technology**, ed. R.W. Rousseau, John Wiley & Sons, NY 1987, 954-981.

Kolff, W.J., and Berk, H.T., The Artificial Kidney: A Dialyzer with a Great Area, Acta Medica Scandinavica, 1944, 117, 121-134.

Lacey, R.E., Handbook of Separation Techniques for Chemical Engineers, ed. P.A. Schweitzer, McGraw-Hill, NY 1988, 1_479-1_495.

Lakshminarayanaiah, N., Transport Phenomena in Membranes, Academic Press, NY & London 1969, 58-361.

Lake, M.A., and Melshedaer, S.S., Mass Transfer Characterization of Donnan Dialysis, American Institute of Chemical Engineers Journal, 1978, 24, 1, 130-137.

Linch, J.D., Acid Recovery Via Dialysis, HPD Incorporated, **Membrane Technology/Planning Conference**, 1985.

Masayuki, O., Uchida, M., and Ishibashi, T., Stainless Steel Pickling by Fixed Concentration HF/HNO₃ Mixtures Through Diffusion Dialysis Acid Reclamation, **Wire Journal International**, 1992, March, 43-48.

McCabe, W.L., Smith, J.C. and Harriton, P., Unit Operations of Chemical Engineering, McGraw-Hill, NY 1993, 860-863.
Mears, P., Synthetic Membranes: Science, Engineering and Applications, ed. P.M. Bungay, H.K. Lonsdale and M.N. de Pinho, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo 1986, 169-179.

Moore, W.J., Physical Chemistry, Third ed., Prentice-Hall Inc., NJ 1962, 273-274.

Nakamura, I., Kawahara, T., and Utsunomiya, T., Use of Dialysis Techniques in Metal Finishing Process, New Materials and New Processes, 1983, 2, 166-175.

Ondrick, E., Use of Diffusion Dialysis to remove Metals from Nickel Plating Solutions at a Nickel Plating Job Shop, New England Water Environment Association Zero Discharge Conference, New Hampshier May 27, 1993.

Perry, R.H., and Green, D.W., Perry's Chemical Engineers' Handbook, 6th ed, McGraw-Hill, NY 1984, 17_18-17_26.

Sato, J., Onuma, M., Motomura, H. and Noma, Y., Recovery of Nitric Acid and Hydrofluoric Acid from the Pickling Solution by Diffusion Dialysis, Tokuyama Soda Co., LTD, Reprint from Jitsumu Hyomen Gijutsu, The metal Finishing Society of Japan 1985, 1-8.

Scott, J., Membrane and Ultrafiltration Technology, Recent Advances, Noyes Data Corporation, NJ 1980, 210-211.

Schindhelm, K., Farrel, P.C., and Stewart, J.H., Convective Mass Transfer in Artificial Kidney, Paper read at 2nd Australian Conference on Heat and Mass Transfer, 16-18 February, 1977, University of Sydney.

Schlögl, R., Berichte der Bunsengesellschaft fuer Physikalische Chemie., 1966, 71, 755.

Simon, G.P., Ion Exchange Training Manual, Van Nostrand Reinhold, NY 1991, 40.

Sridhar, P., and Subramaniam, G., Recovery of Acid From Cation Exchange Resin Regeneration Waste by Diffusion Dialysis, Journal of Membrane Science, 1989, 45, 273-280.

Treybal, R.E., Mass-Transfer Operations, Third ed., McGraw-Hill, NY 1980, 34-36.

Tokuyama Soda CO., LTD., Instruction Manual for Diffusion Dialysis Unit Model TSD-2, Tokuyama Japan 1991.
Tuwiner, S.B., *Diffusion and Membrane Technology*, Reinhold Publishing Corporation, NY & London 1962, 38-60, 77-90, 137-151, 287-320.

Wallace, R.M., Concentration and Separation of Ions by Donnan Membrane Equilibrium, Industrial and Engineering Chemistry, Process Design and Development, 1967, 6, 423-431.

Wendt, R.P., Klein, E., Bressler, H.E., Holand, F.F., Serino, R.M., and Villa, H., Sieving Properties of Hemodialysis Membranes, *Journal of Membrane Science*, 1979, 5, 23-49.

White, F.M., *Fluid Mechanics*, Second ed., McGraw-Hill, NY 1986, 322.

Yasuda, H., Peterlin, A., Colton, C.K., Smith, K.A., and Merril, E.W.
Permeability of Solutes Through Hydrated Polymer Membranes, III.
theoretical background for selectivity of dialysis membranes, *Die
Makromolekulare Chemie*, 1969, 126, 177-186.

Yasuda, H., Lamaze, C.E., and Ikenberry, L.D., Permeability of solutes Through Hydrated Polymer Membranes., I. diffusion of sodium chloride, *Die mMakromolekulare Chemie*, 1968, 118, 19-35.

Yasuda, H., Lamaze, C.E., and Peterlin, A., Diffusive and Hydraulic Permeabilities of Water in Water-swollen Polymer Membranes, *Journal of Polymer Science*, 1971, 9, 1117-1131.