= Solution
Let $a$ denote tube cross-sectional area in this question, not a filament or sphere radius. The bacterial density is uniform across that section in this one-dimensional <Keller--Segel model>, so the total bacterial number is
$$
N=a\int_{-\infty}^{\infty}B(z)\,dz.
$$
Integrating the nutrient balance $vC'=kB$ across the complete band gives
$$
v[C(+\infty)-C(-\infty)]=k\int_{-\infty}^{\infty}B(z)\,dz
=\frac{kN}{a}.
$$
Hence the <speed of a nutrient-consuming chemotactic band> is
$$
\boxed{v=\frac{Nk}{aC_\infty}.}
$$
This is a nutrient budget: the whole population consumes at rate $Nk$, while advancement of the band at speed $v$ brings fresh nutrient at rate $a v C_\infty$. The <conservation of mass> equation for bacteria ensures $N$ stays constant when there is no flux at infinity and no cell birth or death.
At fixed total bacterial number, the speed grows linearly with $N$ and the per-cell consumption rate $k$, and decreases with tube area and nutrient concentration ahead. More nutrient takes longer to deplete, giving the inverse dependence on $C_\infty$. Although $D$ and $\alpha$ determine the width, skewness, peak density and admissibility of the smooth band, they do not appear in this speed when $N,k,a,C_\infty$ are fixed. This independence relies on neglecting nutrient diffusion and using the specified concentration-independent consumption law; it is not a general prediction for all <chemotaxis> models. Dimensionally $Nk$ is nutrient amount per time and $aC_\infty$ nutrient amount per length, so their ratio is a velocity.
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