Page 55 - Entrophy Analysis in Thermal Engineering Systems

The common source of entropy increase 47

Fig. 4.1 Heat generation due to frictional work in a pipe flow.

Goodenough [3]. For an infinitesimal heat δq f (per unit mass of the fluid)
generated at temperature T, the increase of (specific) entropy is

δq
f
ds ¼ (4.5)
T
Consider an incompressible fluid flowing through a horizontal pipe having a
uniform cross-section as sketched in Fig. 4.1. The dashed rectangle shows a
differential element where the temperature is T. The pipe is insulated so
there is no heat exchange between the fluid and its surroundings. The work
done on the left plane of the element is pv, whereas it is pv vdp on the right
plane with p and v denoting the pressure and specific volume of the fluid,
respectively. The amount of heat generated due to the friction equals the
frictional work. Hence,

δq ¼ pv vδpð Þ pv (4.6)
f
Simplifying Eq. (4.6) yields

(4.7)
δq ¼ vδp
f
Eliminating dq f between Eqs. (4.5) and (4.7), one obtains

(4.8)
v
ds ¼ dp
T
The increase of entropy due to the pressure drop between two locations
along the pipe is determined by integrating Eq. (4.8).
ð
dp
(4.9)
Δs ¼ v
T
Eq. (4.9) allows calculation of the entropy increase due to the heat generation
associated with the pressure drop on the flow path of an incompressible fluid.
Note that the analysis rests on the assumption of the entire heat generated
being dissipated to the fluid.