economic aspects depending on the type of vessel and the parameters of
the waterway. It is usually up to 15–20 km/h on a deep and wide river and
11–15 km/h on canals and canalized navigable rivers (class IV).
The resistance of ships on restricted waters is influenced by many
factors, the most important being speed, flow velocity, shape of bow and
stern, length, squat and draught (both at bow and stern), keel clearance,
and distance from canal banks. A general expression for the resistance, R,
of a towed vessel was given by Kaa (1978), in a simplified form, as
RCF (u)^2 A gBTz Cp ^2 BT (11.4)
whereis the ship’s speed, uis the velocity of the return flow (uat the
stern),zis the depression of the water level (equal to the squat), at stern
or bow, CFis the frictional resistance coefficient, Ais the wetted hull area,
Bis the width of the ship, Tis the draught and Cpis a coefficient depend-
ent on speed and draught.
The return flow velocity uand squat zin equation (11.4) can be com-
puted from Bernoulli’s equation and continuity:
2 gz(u)^2 ^2 (11.5)
A(u)(Ac AM ∆Ac) (11.6)
whereAcis the canal cross-section and AMthe midship sectional area. A
good approximation for ∆Acis given by
∆AcBcz (11.7)
whereBcis the undisturbed canal width.
The resistance augmentation for push-tows over a single ship of the
same dimensions and parameters is only moderate.
On a restricted waterway the resistance, R, increases as the ratio, n,
of the canal cross-sectional area Acand the immersed section of the
barge(s)A(total immersed section of a train) decreases approximately in
the ratio of (n/(n 1))^2 (i.e. assuming a variation of Rwith^2 ). It follows
that for values of nless than 4–5 the increase in the resistance becomes
prohibitive (decreasing nfrom 6 to 5 increases Rby 8%, 5 to 4 by 14% and
4 to 3 by 26%) and usually a value of ngreater than 4 is used; (this limit is,
of course, also speed dependent). Figure 11.6 shows the approximate vari-
ation of the resistance Rfor a 1350 t barge as a function of , for three
values of nandT. The curves have been computed from the simple equa-
tion by Gebers (Cˇábelka, 1976):
R(AkBT)2.25 (11.8)
1
2
1
2
474 INLAND WATERWAYS
