List of practice Questions

The theoretical aerobic oxidation of biomass (C₅H₇O₂N) is
\[ \mathrm{C_5H_7O_2N + 5\,O_2 \rightarrow 5\,CO_2 + NH_3 + 2\,H_2O} \] 

Given a first-order biochemical oxidation with $k=0.23\ \text{d}^{-1}$ at $20^{\circ}$C (base $e$) and neglecting the second-stage oxygen demand, find the ratio BOD₅ at $20^{\circ}$C to TOC (round to two decimals). 

Atomic weights: C=12, H=1, O=16, N=14 g/mol

The cross-section of a small river is sub-divided into seven segments of width 1.5 m each. The average depth, and velocity at different depths were measured during a field campaign at the middle of each segment width. The discharge computed by the velocity area method for the given data is m$^3$/s (round off to one decimal place).} 

A catchment may be idealized as a circle of radius 30 km. There are five rain gauges, one at the center of the catchment and four on the boundary (equi-spaced), as shown in the figure (not to scale). The annual rainfall recorded at these gauges in a particular year are given below. 

Using the Thiessen polygon method, what is the average rainfall (in mm, rounded off to two decimal places) over the catchment in that year? 

For the flow setup shown, hydraulic conductivities are $k_1=10$ mm/s (Soil 1) and $k_2=1$ mm/s (Soil 2). Unit weight of water $=10$ kN/m$^3$. Ignore velocity head. At steady state, what is the total head (in m, rounded to two decimals) at the junction of the two samples?

A circular pile of diameter 0.6 m and length 8 m was constructed in a cohesive soil stratum having the following properties: bulk unit weight = 19 kN/m$^3$, angle of internal friction = 0$^\circ$ and cohesion = 25 kPa. The allowable load the pile can carry with a factor of safety of 3 is  kN (round off to one decimal place).}

If the size of the ground area is $6 \, \text{km} \times 3 \, \text{km}$ and the corresponding photo size in the aerial photograph is $30 \, \text{cm} \times 15 \, \text{cm}$, then the scale of the photograph is $1$  

For the elevation and temperature data given in the table, the existing lapse rate in the environment is  °C/100 m (round off to two decimal places).

The pressure in a pipe at X is to be measured by an open manometer as shown in the figure. Fluid A is oil with a specific gravity of 0.8 and Fluid B is mercury with a specific gravity of 13.6. The absolute pressure at X is  kN/m\(^2\). (round off to one decimal place).} 
[Assume Density of water = 1000 kg/m³, gravity = 9.81 m/s², atmospheric pressure = 101.3 kN/m².]

An unconfined compression strength test was conducted on a cohesive soil. The test specimen failed at an axial stress of 76 kPa. The undrained cohesion (in kPa, in integer) of the soil is

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of $100^\circ$C and the other three sides are kept at a temperature of $0^\circ$C. Ignoring the effect of discontinuities at the corners, the steady-state temperature at the center of the plate is obtained as $T_0^\circ$C. Due to symmetry, the steady-state temperature at the center will be the same ($T_0^\circ$C) when any one side of the square is kept at a temperature of $100^\circ$C and the remaining three sides are kept at $0^\circ$C. Using the principle of superposition, find $T_0$ (rounded off to two decimal places).

The wastewater inflow to an activated sludge plant is $0.5\, \text{m}^3/\text{s}$, and the plant is to be operated with a food to microorganism ratio of $0.2\, \text{mg/mg-d}$. The concentration of influent biodegradable organic matter (after primary settling) is $150\, \text{mg/L}$, and the mixed liquor volatile suspended solids (MLVSS) to be maintained is $2000\, \text{mg/L}$. Assuming complete removal of biodegradable organics in the tank, the volume of aeration tank (in m$^3$, integer) required is.

A hydraulic jump occurs in a 1.0 m wide horizontal, frictionless, rectangular channel, with a pre-jump depth of $0.2$ m and a post-jump depth of $1.0$ m. Take g=10 m/s^2. The values of the specific force at the pre-jump and post-jump sections are the same and are equal to (in m$^3$, rounded off to two decimal places)

A 12-hour storm occurs over a catchment and results in a direct runoff depth of 100 mm. The time-distribution of the rainfall intensity is shown in the figure (not to scale). The $\varphi$-index of the storm is (in mm, rounded off to two decimal places): 

A smooth vertical retaining wall supporting layered soils is shown in the figure. According to Rankine's earth pressure theory, the lateral active earth pressure acting at the base of the wall is  kPa (rounded off to one decimal place).

A $5$ cm long metal rod ($x$ in cm from $A$ to $B$) is governed by $\dfrac{\partial T}{\partial t}=D\,\dfrac{\partial^2 T}{\partial x^2}$ with $D=1.0~\text{cm}^2/\text{s}$ and both ends held at $0^\circ$C. The temperature field is \[ T(x,t)=\sum_{n=1,3,5,\ldots} C_n \sin\!\left(\frac{n\pi x}{5}\right)e^{-\beta n^{2} t}. \] Find $\beta$ (in s$^{-1}$, rounded to three decimals). 

Consider the singly reinforced section of a cantilever concrete beam under bending, as shown in the figure (M25 grade concrete, Fe415 grade steel). The stress block parameters for the section at ultimate limit state, as per IS 456: 2000 notations, are given. The ultimate moment of resistance for the section by the Limit State Method is  kN.m (round off to one decimal place).

A 2D thin plate (plane stress) has $E=1.0~\text{N/m}^2$ and Poisson’s ratio $\mu=0.5$. The displacement field is $u=Cx^2y$, $v=0$ (in m). Distances $x,y$ are in m. The stresses are $\sigma_{xx}=40xy~\text{N/m}^2$ and $\tau_{xy}=\alpha x^2~\text{N/m}^2$. Find $\alpha$ (in $\text{N/m}^4$, integer). 

An idealised frame supports a load as shown in the figure. The horizontal component of the force transferred from the horizontal member PQ to the vertical member RS at P is  N (round off to one decimal place).} 

A square footing is to be designed to carry a column load of 500 kN which is resting on a soil stratum having the following average properties: bulk unit weight = 19 kN/m³; angle of internal friction = 0° and cohesion = 25 kPa. Considering the depth of the footing as 1 m and adopting Meyerhof's bearing capacity theory with a factor of safety of 3, the width of the footing (in m) is  (round off to one decimal place)} 
 

A soil having the average properties, bulk unit weight $=19\\{kN/m}^3$, angle of internal friction =25 degree and cohesion $=15\\{kPa}$, is being formed on a rock slope at an inclination of 35 degree with the horizontal. The critical height (in m) of the soil formation up to which it would be stable without failure is (round off to one decimal place). [Assume the soil is formed parallel to the rock bedding plane and there is no ground water effect.] 

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm², in integer) at the point?

The differential equation \(\dfrac{du}{dt} + 2tu^{2} = 1\) is solved by a backward difference scheme. At the \((n-1)\)-th time step, \(u_{n-1}=1.75\) and \(t_{n-1}=3.14\,\text{s}\). With \(\Delta t=0.01\,\text{s}\), find \(u_n-u_{n-1}\) (round off to three decimals).

For a horizontal curve, the radius of a circular curve is 300 m with the design speed 15 m/s. If the allowable jerk is 0.75 m/s$^3$, what is the minimum length (in m, integer) of the transition curve?

The ordinates of a one-hour unit hydrograph (1-hr UH) for a catchment are:

Using superposition, a $D$-hour unit hydrograph is derived. Its ordinates are found to be $3\ \text{m}^3\!/\text{s}$ at $t=1$ hour and $10\ \text{m}^3\!/\text{s}$ at $t=2$ hour. Find the value of $D$ (integer).

A canal supplies water to an area growing wheat over 100 hectares. The duration between the first and last watering is 120 days, and the total depth of water required by the crop is 35 cm. The most intense watering is required over a period of 30 days and requires a total depth of water equal to 12 cm. Assuming precipitation to be negligible and neglecting all losses, the minimum discharge (in m\(^3\)/s, rounded off to three decimal places) in the canal to satisfy the crop requirement is