GATE CE Environmental and Transportation Formula Sheet 2027

By MD ANISH AHAMADUpdated 4 Oct 202610 min read

This sheet collects the Environmental and Transportation Engineering formulas that GATE CE papers set again and again. Each row gives the formula, when it applies and its trap. One-line examples with made-up numbers let you test your recall.

In this guide

Key takeaways
How to read this sheet
Environmental Engineering: water quality and treatment
Environmental Engineering: air pollution and solid waste
Transportation Engineering: sight distance and road curves
Transportation Engineering: railways and runways
Transportation Engineering: traffic flow and signals
Transportation Engineering: pavement design
Transportation Planning: new for 2027, with no formula pattern yet
Quick revision

Key takeaways

Environmental Engineering averages 11.1 marks a paper and Transportation Engineering 10.0, across all sixteen counted papers from 2019 to 2026.
Most Environmental marks come from water and wastewater: BOD, hardness, sedimentation, detention time and the activated sludge ratios.
The BOD rate constant comes in base e or base 10, and mixing them is the commonest slip in the section.
In traffic flow, $q = kv$ holds only with the space-mean speed, which is the harmonic mean of spot speeds.
The summit curve constants are 4.4 for stopping sight distance and 9.6 for overtaking sight distance.
Transportation Planning is new for 2027 and has drawn zero marks so far, so there is no formula pattern to learn yet.

How to read this sheet

log means base 10 and ln means natural log. In road formulas, $V$ is speed in km/h and $v$ is speed in m/s. Divide km/h by 3.6 to get m/s.

Overflow rate is flow per unit surface area of a tank. MLSS is the mixed liquor suspended solids in an aeration tank, in mg/L. Space-mean speed is the average speed over a stretch of road; time-mean speed is the average of speeds at one point. For section marks, see GATE CE subject-wise weightage for every section.

Environmental Engineering: water quality and treatment

BOD, hardness and chlorine

Formula	Symbols / when it applies	Watch out for
$\text{BOD}_t = L_0(1 - e^{-kt})$	$L_0$ ultimate BOD, $k$ base-e rate per day	Base-10 form: $L_0(1 - 10^{-Kt})$ with $K = k/2.303$
$L_t = L_0 e^{-kt}$	BOD still remaining after $t$ days	Remaining is the complement of exerted
$\displaystyle \text{as CaCO}_3 = C \times \dfrac{50}{\text{equivalent weight}}$	Hardness and alkalinity; equivalent weight $= \text{molecular weight}/\text{valency}$	Use equivalent weights, not molecular weights
$\text{Chlorine demand} = \text{dose} - \text{residual}$	All in mg/L	The residual is a reserve, not waste

For $L_0 = 250$ mg/L and $k = 0.2$ per day, $\text{BOD}_5 = 250(1 - e^{-1}) = 158.0$ mg/L. For 48 mg/L of calcium, hardness $= 48 \times 50/20 = 120$ mg/L as $\text{CaCO}_3$ . A dose of 2.0 mg/L leaving 0.3 mg/L gives a demand of 1.7 mg/L.

Two IS 10500 values appear in the book. Total hardness is acceptable up to 200 mg/L, and permissible up to 600 mg/L where no alternative source exists. The minimum free chlorine residual at the tap is 0.2 mg/L.

Trap: A base-e constant of 0.23 per day is a base-10 constant of 0.1 per day. Read which base the question states before you write $e$ or 10.

Sedimentation, detention and activated sludge

Formula	Symbols / when it applies	Watch out for
$\text{SOR} = Q/A_s$	$A_s$ surface area of the tank	Removal depends on area, not depth
$\displaystyle v_s = \dfrac{g(\rho_s - \rho)d^2}{18\mu}$	Stokes' law, laminar settling	Square the diameter; use metres
$t = V/Q$	Theoretical detention time	Convert days to hours at the end
$\displaystyle F/M = \dfrac{Q S_0}{V X}$	$S_0$ influent BOD, $X$ MLSS	Answer is per day
$\displaystyle \text{SVI} = \dfrac{\text{settled volume (mL/L)} \times 1000}{\text{MLSS (mg/L)}}$	Sludge settleability, mL/g	Dropping the 1000 is the classic error

For $Q = 4800$ m³/day on 160 m², the overflow rate is 30 m³/m²/day. A 0.05 mm particle of density 2650 kg/m³ in water with $\mu = 10^{-3}$ Pa·s settles at 2.25 mm/s. A 1500 m³ tank taking 9000 m³/day holds the flow for 4 hours.

For $Q = 4000$ m³/day, $S_0 = 200$ mg/L, $V = 1600$ m³ and $X = 2500$ mg/L, $F/M = 0.2$ per day. A sludge settling to 240 mL/L at an MLSS of 2000 mg/L has an SVI of 120 mL/g, which is good settling.

Sewers and population forecasting

Formula	Symbols / when it applies	Watch out for
$\displaystyle V = \frac{1}{n}R^{2/3}S^{1/2}$ with $R = D/4$	Circular sewer running full	Then check self-cleansing velocity, at least 0.6 m/s at minimum flow
$P_n = P_0(1 + r)^n$	Geometric increase, constant percentage growth	Gives the largest forecast of the three methods

A 0.4 m sewer running full has $R = 0.1$ m. A town of 40,000 growing at 3 per cent a year reaches $40000 \times 1.03^{10} \approx 53{,}757$ in ten years.

Environmental Engineering: air pollution and solid waste

Formula	Symbols / when it applies	Watch out for
$H = h_s + \Delta h$	Stack height plus plume rise	Use $H$ , not $h_s$ , in the plume
$\displaystyle C = \dfrac{Q}{\pi u \sigma_y \sigma_z}\exp\!\left(-\dfrac{H^2}{2\sigma_z^2}\right)$	Ground level, on the plume centreline	$Q$ is emission rate, $u$ wind speed
$\displaystyle v_s = \dfrac{Q}{L \times W}$	Gravity settling chamber	Same overflow-rate logic as a tank
Stability: compare ELR with DALR $= 9.8$ °C/km	ELR above DALR is unstable; below is stable	An inversion is very stable and traps pollutants
$r = \rho_{\text{compacted}}/\rho_{\text{loose}} = V_{\text{loose}}/V_{\text{compacted}}$	Solid waste compaction ratio	Ratio is greater than 1; 0.5 is inverted

A 40 m stack with 20 m of plume rise has $H = 60$ m. With $\sigma_z = 30$ m, the exponential factor is $e^{-2} = 0.135$ . Waste compacted from 250 to 625 kg/m³ has a compaction ratio of 2.5.

Remember: Stability is a comparison, not a number. If the surrounding air cools more slowly than 9.8 °C per km, a rising parcel sinks back and the air is stable.

This sheet is a selection. The book's last-minute sheet of formulae and code constants covers all eight technical sections, with each value traced to a worked solution. It is part of the GATE CE 2027 book.

Transportation Engineering: sight distance and road curves

Sight distances and horizontal curves

Formula	Symbols / when it applies	Watch out for
$\displaystyle \text{SSD} = vt + \dfrac{v^2}{2gf}$	$v$ in m/s, $t$ reaction time, $f$ friction	Convert km/h to m/s first
$\text{OSD} = d_1 + d_2 + d_3$	$d_1 = v_b t$ , $d_2 = v_b T + 2s$ , $d_3 = vT$	Drop $d_3$ on a one-way road
$s = 0.7v_b + 6$ , $T = \sqrt{4s/a}$	$v_b$ overtaken vehicle speed, $a$ acceleration	Use $v_b$ , not the design speed, in $d_1$ and $d_2$
$\displaystyle e + f = \dfrac{V^2}{127R}$	$V$ in km/h, $R$ in m	IRC caps $e$ at 0.07 in plain terrain; $f$ at 0.15
$\displaystyle L_s = \dfrac{0.0215V^3}{CR}$ , $\displaystyle C = \dfrac{80}{75 + V}$	Transition length, rate of change of centrifugal acceleration	$V$ in km/h, not m/s

At 72 km/h ( $v = 20$ m/s), with $t = 2.5$ s and $f = 0.35$ , the SSD is $50 + 58.25 = 108.25$ m. At $V = 60$ km/h and $R = 200$ m, $e + f = 0.142$ . With $e = 0.07$ , the friction needed is 0.072, inside the limit. There, $C = 0.593$ and $L_s = 39.18$ m.

Summit and valley curves

Formula	Symbols / when it applies	Watch out for
$\displaystyle L = \dfrac{NS^2}{\left(\sqrt{2h_1} + \sqrt{2h_2}\right)^2}$	Summit, $L > S$ ; $N$ deviation angle	Constant 4.4 for SSD ( $h_1 = 1.2$ m, $h_2 = 0.15$ m); 9.6 for OSD
$\displaystyle L = \dfrac{NS^2}{2(h_1 + S\tan\alpha)}$	Valley, headlight sight distance, $L > S$	Not the summit formula; check comfort too

For $N = 0.06$ and an SSD of 120 m, a summit curve needs $L = 0.06 \times 120^2/4.4 = 196.4$ m. For a valley with $N = 0.06$ , $S = 100$ m, $h_1 = 0.75$ m and $\alpha = 1^\circ$ , $L = 120.2$ m.

In one line: Summit curves are designed on sight distance alone; valley curves take the greater of headlight distance and comfort.

Transportation Engineering: railways and runways

Formula	Symbols / when it applies	Watch out for
$\displaystyle e = \dfrac{GV^2}{127R}$	Equilibrium cant; $G$ gauge in m, $V$ in km/h	Broad gauge maximum cant is 165 mm
Elevation correction: $+7$ per cent per 300 m	Above mean sea level, on the basic runway length	Apply temperature correction to the elevation-corrected length

On broad gauge ( $G = 1.676$ m) at 60 km/h and $R = 400$ m, $e = 0.1188$ m, or 118.8 mm. A 1800 m basic runway at 600 m elevation becomes $1800 + 1800 \times 0.07 \times 2 = 2052$ m.

Transportation Engineering: traffic flow and signals

Formula	Symbols / when it applies	Watch out for
$q = k v_s$	Flow, density, space-mean speed	Not the time-mean speed
$\displaystyle v_s = \dfrac{n}{\sum 1/v_i}$ ; $\displaystyle v_t = \dfrac{\sum v_i}{n}$	Harmonic and arithmetic means of spot speeds	$v_s \le v_t$ always
$v = v_f(1 - k/k_j)$ , $\displaystyle q_{\max} = \dfrac{v_f k_j}{4}$	Greenshields linear model	Capacity at $k_j/2$ and $v_f/2$
$\displaystyle \text{PHF} = \dfrac{\text{hourly volume}}{4 \times \text{peak 15-min volume}}$	Uniformity of the peak hour	Never above 1.0
$\displaystyle C_0 = \dfrac{1.5L + 5}{1 - Y}$	Webster; $L$ lost time per cycle, $Y$ sum of critical flow ratios	$Y$ near 1 makes the cycle explode
$c = s \times g/C$	Approach capacity; $s$ saturation flow	Saturation flow alone is not capacity

For spot speeds of 30 and 60 km/h, $v_t = 45$ km/h but $v_s = 40$ km/h. With $v_f = 90$ km/h and $k_j = 120$ veh/km, $q_{\max} = 2700$ veh/h. An hour of 2400 vehicles peaking at 720 in 15 minutes gives a PHF of 0.833. Webster with $L = 10$ s and $Y = 0.6$ gives a 50 s cycle.

Trap: At a signal, 1800 veh/h is a rate during green. With 30 s of green in a 90 s cycle, the approach carries 600 veh/h.

Transportation Engineering: pavement design

Formula	Symbols / when it applies	Watch out for
$\displaystyle N = \dfrac{365 \times A \times D \times F \times [(1 + r)^n - 1]}{r}$	IRC:37 design traffic, million standard axles; $A$ commercial vehicles/day, $D$ lane distribution, $F$ VDF	Divide by $10^6$ for msa
$\displaystyle \text{damage} \propto \left(\dfrac{\text{axle load}}{\text{standard axle}}\right)^4$	Fourth-power law; standard axle 80 kN	Twice the load does sixteen times the damage
$\displaystyle l = \left[\dfrac{Eh^3}{12(1 - \mu^2)k}\right]^{1/4}$	Westergaard radius of relative stiffness, IRC:58 rigid pavements	Fourth root, so $l$ grows as $h^{3/4}$
$\displaystyle \text{CBR} = \dfrac{\text{test load}}{\text{standard load}} \times 100$	Standard load 1370 kg at 2.5 mm	Take the 2.5 mm value unless 5 mm is higher

With $A = 2000$ , $D = 0.75$ , $F = 3$ , $r = 0.05$ and $n = 10$ years, the growth series is 12.578 and $N = 20.66$ msa. An axle 1.5 times the standard does $1.5^4 = 5.06$ times the damage. For $E = 3 \times 10^5$ kg/cm², $h = 25$ cm, $\mu = 0.15$ and $k = 6$ kg/cm³, $l = 90.34$ cm. A load of 68.5 kg at 2.5 mm gives a CBR of 5 per cent.

Transportation Planning: new for 2027, with no formula pattern yet

The 2027 syllabus adds four-step travel demand modelling. The steps run in order: trip generation, trip distribution, mode choice and traffic assignment. Each answers one question: how many trips, where to, by what mode, and by which route.

It has drawn zero marks in all 1,040 counted questions from the sixteen papers of 2019 to 2026. There is no pattern to learn from, so read the four steps once. The GATE CE 2027 syllabus changes guide lists every addition in both sections.

More formula sheets: all of GATE Civil · Fluid Mechanics and Hydrology · Geotechnical · Structural Engineering

Quick revision

BOD exerted is $L_0(1 - e^{-kt})$ ; switch to base 10 only with $K = k/2.303$ .
Convert hardness with equivalent weights; IS 10500 total hardness is 200 mg/L acceptable, 600 permissible.
Overflow rate is $Q/A_s$ , F/M is $QS_0/(VX)$ per day, and SVI needs the factor of 1000.
Use the effective stack height $h_s + \Delta h$ in the plume, and compare ELR with 9.8 °C/km for stability.
SSD needs speed in m/s; $e + f = V^2/(127R)$ needs km/h, with $e$ capped at 0.07.
Summit curves use 4.4 for SSD and 9.6 for OSD; broad gauge cant stops at 165 mm.
$q = kv$ takes the space-mean speed; Greenshields capacity is $v_f k_j/4$ ; Webster is $(1.5L + 5)/(1 - Y)$ .
Check your calculator steps against the virtual calculator guide, and fit this sheet into the last two months plan.

Frequently asked questions

Is the BOD rate constant in base e or base 10?

Both conventions exist, so read the question. In base e the BOD exerted is $L_0(1 - e^{-kt})$ . In base 10 it is $L_0(1 - 10^{-Kt})$ , with $K = k/2.303$ . A base-e constant of 0.23 per day equals a base-10 constant of 0.1 per day. Using 0.23 in the base-10 form inflates the answer, and that inflated value is the standard wrong option.

Why does the relation

q = kv

need the space-mean speed?

Flow equals density times speed only when the speed is averaged over a length of road, which is the space-mean speed. Space-mean speed is the harmonic mean of spot speeds and is never larger than the time-mean speed, which is the arithmetic mean. Using the time-mean speed in $q = kv$ overestimates the flow.

Should I use 4.4 or 9.6 in the summit curve formula?

Use 4.4 when the curve is designed for stopping sight distance, with an eye height of 1.2 m and an object height of 0.15 m. Use 9.6 for overtaking sight distance, where both heights are 1.2 m. Both constants belong to the case where the curve is longer than the sight distance, so check that $L > S$ after you compute.

Is Transportation Planning examined in GATE CE?

It is new in the 2027 syllabus as four-step travel demand modelling: trip generation, trip distribution, mode choice and traffic assignment. It drew zero marks in all 1,040 counted questions from 2019 to 2026, so it is untested in the counted record. There is no past pattern to learn from, so read the four steps once and know what each one answers.

Can I take a formula sheet into the GATE CE exam?

No paper of your own is normally allowed in the hall. The exam gives you a virtual calculator and a scribble pad. Many candidates write a few easily forgotten constants, such as 4.4, 9.6 and 0.0215, on the pad in the first minutes. The rules can change by year, so confirm the current rules at gate2027.iitm.ac.in.

Sources

Dates, fees and the syllabus are set by the GATE 2027 organising institute and can change. Always confirm at gate2027.iitm.ac.in.