Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.7 SHAFTWORK 82

#

System A

water

#

System B

Figure 3.10 Two systems with shaft work. The dashed rectangles indicate the system

boundaries. System A has an internal weight, cord, and pulley wheel in air; system B

has a stirrer immersed in water.

The two systems shown in Fig.3.10will be used to illustrate two different kinds of shaft
work. Both systems have a straight cylindrical shaft passing through the system boundary.
Let#be the angle of rotation of the shaft in radians, and!be the angular velocity d#=dt.
Tangential forces imposed on one of these shafts can create a torquesysat the lower end
within the system, and a torquesurat the upper end in the surroundings.^15 The sign con-
vention for a torque is that a positive value corresponds to tangential forces in the rotational
direction in which the shaft turns as#increases.
The condition for!to be zero, or finite and constant (i.e., no angular acceleration), is
that the algebraic sum of the imposed torques be zero:sysDsur. Under these conditions
of constant!, the torque couple creates rotational shear forces in the circular cross section
of the shaft where it passes through the boundary. These shear forces are described by
aninternal torquewith the same magnitude assysandsur. Applying the condition for
zero angular acceleration to just the part of the shaft within the system, we find thatsysis
balanced by the internal torquebexerted on this part of the shaft by the part of the shaft in
the surroundings:bDsys. The shaft work is then given by the formula

wD

Z# 2

# 1

bd#D

Z# 2

# 1

sysd# (3.7.1)

(shaft work, constant!)

In system A of Fig.3.10, when!is zero the torquesysis due to the tension in the cord
from the weight of massm, and is finite:sysD mgrwhereris the radius of the shaft
at the point where the cord is attached. When!is finite and constant, frictional forces at
the shaft and pulley bearings makesysmore negative thanmgrif!is positive, and less
negative thanmgrif!is negative. Figure3.11(a) on the next page shows how the shaft
work given by Eq.3.7.1depends on the angular velocity for a fixed value ofj# 2 # 1 j. The
variation ofwwith!is due to the frictional forces. System A has finite, reversible shaft
work in the limit of infinite slowness (!! 0 ) given bywDmgrÅ#. The shaft work is
least positive or most negative in the reversible limit.

(^14) To prove this, we writem.dv=dt/dzDm.dz=dt/dvDmvdvDd
 1
2 mv
2 DdEk.
(^15) A torque is a moment of tangential force with dimensions of force times distance.