Page 347 - Bird R.B. Transport phenomena

Problems 331

(a) Show that the differential equations describing the viscous flow and heat conduction may
be written in the forms

in which ф = v /v , £ = x/b, and Br = ti ovl/k oT o (the Brinkman number).
z b
(b) The equation for the dimensionless velocity distribution may be integrated once to give
d(j>/dt; = C } • (<p/(p 0), in which С г is an integration constant. This expression is then substituted
into the heat balance equation to get

^7) + BrCftl + ft© + p ® + ) = О (10С.2-5)
2
Obtain the first two terms of a solution in the form
2
©(£ Br) = Br©,(0 + Br 0 2(f) + • • • (10С.2-6)
2
ф(£ Br) = ф + Bnfo(0 + Вг ф © + • • • (10C.2-7)
0 2
It is further suggested that the constant of integration Q also be expanded as a power series
in the Brinkman number, thus
2
QCBr) = C + BrC u + Br C + • • • (10C.2-8)
12
10
(c) Repeat the problem, changing the boundary condition at у = b to q x = 0 (instead of speci-
fying the temperature). 4
3
Answers: (b) ф = f - ^BrfrCf - 3f 2 + 2f ) + • • •
0 = ^
(с) Ф = t -
0 =
10C.3. Viscous heating in a cone-and-plate viscometer. 5 In Eq. 2B.11-3 there is an expression for
the torque 2Г required to maintain an angular velocity П in a cone-and-plate viscometer
with included angle ф (see Fig. 2B.11). It is desired to obtain a correction factor to account
0
for the change in torque caused by the change in viscosity resulting from viscous heating.
This effect can be a disturbing factor in viscometric measurements, causing errors as large
as 20%.
(a) Adapt the result of Problem 10C.2 to the cone-and-plate system as was done in Problem
2B.ll(a). The boundary condition of zero heat flux at the cone surface seems to be more realis-
tic than the assumption that the cone and plate temperatures are the same, inasmuch as the
plate is thermostatted and the cone is not.
(b) Show that this leads to the following modification of Eq. 2B.11-3:

T z =

2
2
where Br = /л п Я /к Т 0 is the Brinkman number. The symbol /x stands for the viscosity at
0
0
0
the temperature T .
o
4 R. M. Turian and R. B. Bird, Chem. Eng. ScL, 18, 689-696 (1963).
5
R. M. Turian, Chem. Eng. Sci., 20, 771-781 (1965); the viscous heating correction for non-Newtonian
fluids is discussed in this publication (see also R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of
Polymeric Liquids, Vol. 1, 2nd edition, Wiley-Interscience, New York (1987), pp. 223-227.