Cover Page for Precalculations – Individual Portion
Laminar Flow Viscometer
Prepared by Professor J. M. Cimbala, Penn State University
Latest revision: 11 January 2012
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________________________________________
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ME 325._____
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Precalculations
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Precalculations
Consider the quasi-steady, incompressible energy equation for flow from the reservoir surface, through the sudden
contraction and ball valve, and through the capillary tube, as sketched in Figure 1 of the Equipment section. The term “quasisteady” refers to the assumption that the liquid level in the reservoir is essentially constant, thereby eliminating all of the
time-dependent terms in the full form of the energy equation. Also, we recognize that the inlet and/or outlet may not be onedimensional, and thus, the kinetic energy correction factor  is included to account for the non-uniform velocity profile. For
the case of one inlet and one outlet, the steady, incompressible energy equation in head form is given as follows:
P1
g

1
V12
2g

z1

Wpump
mg

Wviscous
mg

P2
g
2

V2 2
2g
 z2

Wturbine
mg
 hL, total (1)
(4)
1.
List all the assumptions that can be made to simplify Eq. (1) for our particular problem:
(3)
2.
Cross off the appropriate terms in Eq. (1), justifying each simplification in the space below the equation. Then
write the simplified energy equation as Eq. (2) below:
(2)
(5)
3.
From z1 to z2, list the major and minor losses which contribute to the total irreversible head loss, hL, total (see
Figure 1).
Major losses:
Minor losses:
Assume that the minor losses are negligible with respect to the major loss through the long straight capillary tube
of diameter d. With this assumption, the head loss is defined by
2
L V2
d 2g
where, assuming that the flow remains laminar, the friction factor f for a round tube of diameter d is
64
f 
Red
hL, total   hmajor  f
(3)
(4)
(3)
4.
Combine Eq's. (3) and (4), using the definition of Reynolds number, to express the coefficient of viscosity  as an
explicit function of head loss hL, total. Show all of your work in the space below, and write your equation for  as
Eq. (5) in the space provided.

(3)
5.
Combine Eq's. (2) and (5) to solve for viscosity  as an explicit function of (z1 - z2), V2, and the parameters , g, d,
2, and L. Show all your work in the space below, and write your result as Eq. (6).

To eliminate the velocity in the above equation, apply the definition of volumetric flow rate,
d2
Q  V  V2 A2  V2
4
(5)
6.
(5)
(6)
(7)
Substitution of Eq. (7) into Eq. (6) yields the final formula for  in terms of measurable parameters. Showing all
your work in the space below, write this as Eq. (8):

(8)
Write down Eq. (8) on a separate sheet of paper, and bring it with you to the lab. (This is necessary since you need the
equation to perform the lab, but you will hand in these Precalculations before starting the lab.)
(2)
7.
Before proceeding, verify your Eq. (8) by plugging in the following example numbers:

=
g
=
(z1 - z2) =
2
=
997 kg/m3
9.81 m/s2
0.372 m
2.0 (for laminar pipe flow)
d
L
V (= Q)
=
=
=
0.00104 m
0.229 m
0.482 mL/s
With the above example numbers, your Eq. (8) should yield  = 8.63  104 Ns/m2. Verify this in the space
provided below. Be sure to write down all the units, and make sure they reduce to the appropriate units for . Use
consistent SI units everywhere.
Sample  =
This completes the analysis. The viscosity which you desire to measure is now related in a simple manner to
easily controlled or measurable parameters in Eq. (8) – subject, of course, to the assumptions inherent in your
analysis.
(5)
8.
In the space below, explain which capillary tube(s) should yield the most accurate viscosity measurements. In
particular, is it better to use a short tube or a long tube? Why? Is it better to use a small diameter tube or a large
diameter tube? Why? Hint: The goal is to minimize the minor losses with respect to the major losses so that Eq.
(8), in which minor losses were neglected, is as accurate as possible.
Experimental Objectives
a.
b.
c.
d.
Design a procedure to measure the viscosity of water.
Measure the viscosity of water at room temperature using the procedure above. Several different tubes will be
used to determine the tube which yields the most accurate viscosity data.
Measure the viscosity of water at several temperatures.
Compare results with viscosity data from the literature.