Physical Chemistry Third Edition

456 10 Transport Processes

Exercise 10.6

The inside diameter of the buret in Example 10.6 is equal to 0.80 cm. Calculate the speed of the

meniscus when the meniscus is at the 0.00-mL mark.

EXAMPLE10.7

The blood pressure difference across a capillary in a human body is approximately 22 torr.

Assume that a human body contains 1× 1010 capillaries with an average length of 8× 10 −^4 m

and an average diameter of 7× 10 −^6 m. Although blood is a non-Newtonian fluid and contains

red blood cells with diameter near 7× 10 −^6 m, assume that a Newtonian viscosity of 0.004 kg

m−^1 s−^1 can be used (“Blood is thicker than water”). Estimate the volume of blood flowing

through the human circulatory system in 1.00 minute, assuming Poiseuille’s equation and

assuming that the resistance to flow in the arteries and veins is negligible compared with that

in the capillaries. The actual volume is approximately 5 liter per minute, which corresponds

roughly to the entire volume of blood in the body circulating each minute.

Solution

dV

dt



(P 2 −P1)πR^4

8 Lη

×(number of capillaries)



(22 torr)

(

101325 Pa

760 torr

)

π(3. 5 × 10 −^6 m)^4

8(8× 10 −^4 m)(0.004 kg m−^1 s−^1 )

(1× 1010 ) 5. 4 × 10 −^4 m^3 s−^1

(5. 4 × 10 −^4 m^3 s−^1 )

(

60 s

1 min

)(

1 min

1m^3

)

32 L min−^1

This value is too large by a factor of 6, which is not surprising because the red blood cells

have a diameter roughly equal to the capillary diameter and the flow cannot be assumed to

be laminar.

If the flow of a fluid is turbulent the problem is much more complicated and we

will not attempt to discuss it. There is a dimensionless quantity called theReynolds

numberthat can be used to determine whether flow through a tube is likely lami-

nar. The Reynolds number is denoted byRand is defined for flow in a cylindrical

tube by

R

R〈u〉ρ

η

(definition of Reynolds number) (10.2-26)

whereRis the radius of the tube,ρis the density of the fluid,ηis the viscosity of

the fluid, and〈u〉is the mean speed of flow in the tube. It is found experimentally

that flow in a tube is probably laminar if the Reynolds number is smaller than 2000,

no matter what the values of the individual quantities in Eq. (10.2-26) are. If the tube

is long, smooth, and straight, the flow might be laminar if the Reynolds number is

as large as 3000, but it is best to assume that the flow is not laminar ifRexceeds

2000. If calculation indicates that a flow is not laminar, Poiseuille’s equation cannot
      be used.