Q$2.$ A culvert is to be designed and constructed with a return period of $50$ years at a point on a stream.

Given below are the maximum flow records at a point where a drainage structure is to be designed.

a$)$ Using Weibul's formula given below, determine the $50$ year flood

$12$ marks

$T_r=\frac{n+1}{m}$

$m-$ rank of the event; $n-$ total number of event

b$)$ The average difference in elevation between the river bed and finished level of the road at the culvert

location is $3\mathrm{.}75m.$ The length of the embankment is $65m,$ slope is $1\mathrm{.}5\%$ and the pool elevation at the outlet

at $50$ year return period is $1\mathrm{.}5m$ above the invert. Use a free board of $0\mathrm{.}5m,f=0\mathrm{.}016$ for concrete, $k_{en}=0\mathrm{.}5,$

and $k_{ex}=1\mathrm{.}0$

Design an adequately sized concrete pipe culvert of square edge with head wall for the road for the

$50-$ year return period flood and indicate the condition of operation of the culvert for the design flood. Q$2.$ A concrete culvert is to be provided at a point on a river course over which a road is to be constructed.

The catchment area of the river above the culvert location is $1400$ hectares and its stream length is $25km.$

The fall in level of the catchment from the farthest point to the culvert site was found to be $170m.$ The

vegetation cover of the catchment is found to be varied and shown in Table $1$ below

Table $1$: Vegetation cover and runoff coefficient for the catchment

The Rainfall Intensity characteristics for the catchment for the various return periods are represented by the

constants $a,b,c$ in Table $2$ from which the design intensity can be calculated using the equation

$i_n=\frac{a}{(t_c+b{)}^c}$

Where

$i_n=$ rainfall intensity for a given return period, $n(m\frac{m}{h}r)$

$t_c=$ time $of$ concentration $(hrs)$

$a,b,c=$ constants given $in$ Table $2$

Table $2.$ Rainfall Intensity $-$ Return Period Constants for Catchment

The area reduction factors $($ARF$)$ for the catchment are given by the Table $3.$

Table $3.$ ARF for the catchment

a$)$ Calculate the time of concentration $(t_c)$ using

i$)$ Kirpich formula:

$t_c=(\frac{0\mathrm{.}87L^2}{1000S}{)}^{0\mathrm{.}385}$

Where;

$t_c=$ time $of$ concentration $(hrs)$

$L=$ stream length $(km)$

$S=$ mean slope $of$ stream $(\frac{m}{m})$

ii$)$ Bransby Williams formula:

$t_c=\frac{0\mathrm{.}96L^{1\mathrm{.}2}}{H^{0\mathrm{.}2}{A}^{0\mathrm{.}1}}$

Where;

$t_c=$ time $of$ concentration $in(hrs)$

$L=$ stream length $(km)$

$H=$ Elevation change $(m)$

$A=$ catchment area $(K{m}^2)$

b$)$ Given a design period of $25$ years, calculate the peak design runoff using the two $(2)$ times of

concentration you have calculated. Comment on the different discharge values $($ff any$).$

$18$ marks

From the discharge estimated by the Bransby Williams formula, Design an adequately sized concrete

pipe culvert to pass the $25$ years flood. The height of the road above the invert level of the river at

the culvert location is $2\mathrm{.}85m,$ the length of the culvert is $60m,$ natural streambed slope $=0\mathrm{.}01$ and

the downstream level of the $25$ years flood is $1\mathrm{.}1m$ above the downstream invert. Use a groove end

with headwall with $k_e=0\mathrm{.}2$;$n$ for concrete $=0\mathrm{.}015$;$f=0\mathrm{.}02.$ You may consider multiple culverts if the

diameter of pipe becomes too big to be closer to headwater.

Determine the conditions of flow in the culvert by determining: i$)$ the critical depth and ii$)$ the normal

depth using the charts attached.

c$)$ Because of the easily erodible nature of the soil at the culvert outlet, it is necessary to protect the

downstream side from excessive erosiob$).$ An urban drainage consists of an underground pipe of diameter $d=0\mathrm{.}9m$ laid on a slope of $0\mathrm{.}01$ and

discharges the flood discharge at a depth of $0\mathrm{.}9d.$ Somewhere along the flow, the slope changed bringing

the depth in the drain to $0\mathrm{.}7d.$ If the Manning's roughness coefficient of the pipe is $n=0\mathrm{.}013,$ what is the

magnitude of the flood discharge.

$11$ marks

Determine also: i$)$ the critical depth corresponding to this flow using the attached nomogram. $2$ marks

ii$)$ The slope in the second section of the drain

iii$)$ The Froude Number of flow for the two sections of the drain

iv$)$ The type of flow in both sections

v$)$ Explain what will be the form of the fluid surface in the region of the change of slope.