## 3 Some Simple One-Dimensional Transport Scenarios
### Frictional Transport Scenarios  
The transport scenarios discussed above still represent highly idealized versions in that these completely neglect the effects of friction. In a friction-dominated transport scenario, acceleration and deceleration processes are the result of an interplay of forward-driving  
Figure 7: Transport value (blue) and energy relaxation time (red) as a function of the relative duration \(\delta/\tau_{AB}\) of acceleration and deceleration phases. Both functions are measured relative to the physical action \(W_{AB}(M,L_{AB},\tau_{AB})\) and the time duration \(\tau_{AB}\) of an ideal “fly-through” transport connecting start- and endpoints \(A\) and \(B\) over a distance \(L_{AB}\). engine (\(F_{en}\)) and retarding frictional forces (\(F_{fr}\)), giving rise to a stationary velocity \(v_{AB}\) (Figure 8a):  
\[M\frac{dv(t)}{dt}=F_{en}-F_{fr}=0 \tag{11}\]  
In road-traffic situations, cars usually experience frictional forces that scale quadratically with the momentary speed \(v(t)\)[23; 24]:  
\[F_{fr}(t)=\frac{1}{2}\varrho_{air}\ A\ C_{d}v(t)^{2}. \tag{12}\]  
In this equation, \(v(t)\) is its momentary velocity, \(\varrho_{air}\) the air mass density, \(C_{d}\) the dimensionless air drag resistance, and \(A\) the cross-sectional area of the vehicle. Considering long transport distances \(L_{AB}\) in which acceleration and deceleration phases close to start-points \(A\) and endpoints \(B\) do not play significant roles, engine power \(P_{en}=F_{en}v_{AB}\) needs to be constantly applied to overcome the effect of air friction and to maintain a constant speed \(v_{AB}\), as shown below:  
\[P_{en}(v_{AB})=\frac{1}{2}\varrho_{air}\ A\ C_{d}v_{AB}^{3}. \tag{13}\]  
Double integration over the transport time \(\tau_{AB}\) first yields the energy input that is required for maintaining the speed \(v_{AB}\) over the entire transport length \(L_{AB}\) and then the physical action that is associated with the frictional transport over the length \(L_{AB}\). With \(\tau_{AB}=L_{AB}/v_{AB}\), the total energy input along the transport length \(L_{AB}\) emerges as  
\[E_{fr}(L_{AB},v_{AB})=\frac{1}{2}\varrho_{air}\ A\ C_{d}v_{AB}^{2}L_{AB}, \tag{14}\]  
and the generated transport value as  
\[W_{fr}(L_{AB},v_{AB})=\frac{1}{4}\varrho_{air}\ A\ C_{d}v_{AB}L_{AB}^{2}. \tag{15}\]  
Figure 8: (**a**): Start–stop scenario with \(\delta=0\) and frictional forces acting along the transport path \(L=\overset{\rightarrow}{AB}\). Arrows denote the direction of the transport (red), the inputs in engine energy (blue) and the withdrawal of energy through air friction and braking (brown). Energy relaxation times \(\tau_{fr}\) in friction-limited transport scenarios are normally in the order of several minutes [23; 24]. (**b**): Left-hand ordinate: Physical action versus distance attained for frictional transport (\(W_{fr}\)), and frictionless “fly-through” transport (\(W_{AB}\)), all plotted versus transport length. Right-hand ordinate: transport value of frictional transport relative to frictionless “fly-through” transport versus transport length. In the evaluation, a constant speed of \(v_{AB}=100\) km/h is assumed. The crossover points at \(L_{AB}\cong 4\) km are only weakly dependent on \(v_{AB}\). With Equations (14) and (15), the energy efficiency of the frictional transport becomes  
\[\eta_{E\_\_{rf}}(M,L_{AB},\tau_{AB})=\frac{W_{fr}(L_{AB},\upsilon_{AB})}{E_{fr}( L_{AB},\upsilon_{AB})}=\frac{\tau_{AB}}{2}, \tag{16}\]  
which shows that air friction reduces the energy relaxation time \(\tau_{E\_\_{rf}}\) to a much shorter value than in the frictionless start stop-scenario with \(\delta=0\), where it remained at \(\eta_{E\_AB}(M,L_{AB},\tau_{AB})=\tau_{AB}\).