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Chapter 1
Solute Transport In Biological Systems
Example 1.3-1. ---------------------------------------------------------------------------------An experimental reverse osmosis membrane is being used to produce a waste stream that is
Na2CO3 solution. The pure water flux at 30oC is measured as Jsolvent = 1500 L/m2day when
p = 25 atm. With the Na2CO3 solution (5 wt% Na2CO3) operation is at 30oC with PH = 50
atm and PL = 1 atm. Both permeate and feed sides of the membrane are perfectly mixed. Find
Jsolvent for the 5 wt% Na2CO3 solution. The vapor pressure of water over 5 wt% Na2CO3
solution is 31.2 mmHg.
Solution ------------------------------------------------------------------------------------------
Membrane
1 atm
50 atm
Pfeed-Ppermeate
Water
5 wt% Na2CO3
feed-permeate)
Low pressure
(permeate) side
High pressure
(feed) side
Figure 1.3-5 Flow of water by reverse osmosis.
The solvent volumetric flow rate is given by
Jsolvent = LpS[(Pfeed  Ppermeate)  (feed  permeate)]
We can determine the product LpS from pure water flow data when (feed  permeate) = 0
LpS =
J solvent
1500
=
= 60 L/m2dayatm
25
Pfeed  Ppermeate
When the feed side is 5 wt% Na2CO3 solution, the osmotic pressure must be determined. We
will consider two cases: ideal and non-ideal solution.
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1) For ideal solution, the osmotic pressure on the feed side is given by
 = RT CS
Density of Na2CO3 is 2.533 g/cm3, and density of water is 1 g/cm3. For 1000 g of 5 wt%
Na2CO3 solution we have 950 g of water and 50 g of Na2CO3. Since the molecular weight of
Na2CO3 is 106, CS is computed as
CS =
50 / 106
= 4.8610-4 mol/cm3
950 / 1  50 / 2.533
With the ideal gas constant R = 82.057 cm3atm/moloK, the osmotic pressure for the feed
side is
 = (82.057)(30 + 273.15)(4.8610-4) = 12.1 atm
The solvent flow rate is then determined
Jsolvent = LpS[(Pfeed  Ppermeate)  (feed  permeate)]
Pfeed  Ppermeate = 50  1 = 49 atm
feed  permeate = 12.1  0 = 12.1 atm
Jsolvent = 60[49  12.1] = 2814 L/m2day
2) For non-ideal solution, the osmotic pressure on the feed side is given by
= 
RT
ln  W xW 
L
VW
The activity coefficient  W in 5 wt% Na2CO3 solution can be obtained from the requirement
that the fugacity of water in the solution is the same as the fugacity of water vapor in
equilibrium with the solution
f WL = f WV  xW  W PWo = yWP = PW  xW  W =
PW
PWo
At 30oC, vapor pressure of water is PWo = 31.824 mmHg, vapor pressure of 5 wt% Na2CO3
L
solution is PW = 31.2 mmHg, and V W = 18.095 cm3/mol. Therefore
= 
P
(82.057)( 303.15)
31.2
RT
ln Wo = 
ln
= 27.2 atm
L
18.095
31.824
PW
VW
The solvent flow rate is then
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Jsolvent = LpS[(Pfeed  Ppermeate)  (feed  permeate)]
Jsolvent = 60[49 – 27.2] = 1307 L/m2day
The assumption of ideal solution can produce a significant error.
--------------------------------------------------------------------------------------------------Table 1.2-1 lists the osmotic pressure of aqueous sucrose solution at 30oC calculated using
equation (1.2-3) for non-ideal solution and equation (1.2-5) for ideal solution. The values for
non-ideal model agree well with the experimental data.
RT
ln  WA xWA 
L
VW
(1.2-3)
RT A
x S = RT CS
L
VW
(1.2-5)
= 
=
Table 1.2-1 Osmotic pressure (atm) of aqueous sucrose solution at 30oC
CS (mol/liter)
Eq. (1.2-5)
Eq. (1.2-3)
Exp. Data
0.991
20.3
26.8
27.2
1.646
30.3
47.3
47.5
2.366
39.0
72.6
72.5
3.263
47.8
107.6
105.9
4.108
54.2
143.3
144.0
5.332
61.5
199.0
204.3
1.4 Regulation of Extracellular Fluid Osmolality1
Water that is consumed is absorbed from the intestines and enters the extracellular fluid. An
increase in the osmolality of the extracellular fluid triggers thirst and antidiuretic hormone
(ADH) secretion. ADH acts on the distal tubules and collecting ducts of the kidneys to
increase reabsorption of water from the filtrate. The increase in the amount of water entering
the extracellular fluid causes a decrease in osmolality.
A decrease in extracellular fluid osmolality inhibits thirst and ADH secretion. Less water is
consumed, and less water is reabsorbed from the filtrate in the kidney. Consequently, more
water is lost as a large volume of dilute urine. The result is an increase in the osmolality of
the extracellular fluid.
The ADH and thirst mechanism are sensitive to even small changes in extracellular fluid
osmolality and the response in fast (from minutes to a few hours). The hormonal regulation
of blood osmolality is illustrated in Figure 1.4-1. The diencephalon, part of the brain between
the brainstem and the cerebrum, is shown in Figure 1.4-2. Its main components are the
thalamus, subthalamus, epithalamus, and hypothalamus.
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Seeley R.R, Stephens T.D., Tate P., Anatomy & Physiology, McGraw Hill, 2003, p. 989
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Blood osmolality
(normal range)
An increase in blood osmolality is detected by
osmoreceptors in the hypothalamus
An decrease in blood osmolality is detected by
osmoreceptors in the hypothalamus
Osmoreceptors stimulate ADH secretion from
the posterior pituitary and increase thirst
Osmoreceptors inhibit ADH secretion from
the posterior pituitary and decrease thirst
* Increased ADH increases the permeability of
the distal tubules and collecting ducts to
water. More water returns to the blood and
less water is lost in the urine.
* Decreased ADH decreases the permeability of
the distal tubules and collecting ducts to
water. Less water returns to the blood and
more water is lost in the urine.
* Increased thirst increases water intake,
resulting in the increased movement of water
into the blood
* Decreased thirst decreases water intake,
resulting in the decreased movement of water
into the blood.
A decrease in blood osmolality results from the
increased movement of water into the blood
An increase in blood osmolality results from the
decreased movement of water into the blood
Blood osmolality
(normal range)
Figure 1.4-1 Hormonal regulation of blood osmolality.
Figure 1.4-2 A graphical depiction of the diencephalon.
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