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Applied Field Engineering: formula sheet

Ground, air, water & what it costs · 122 formulas · metric edition 1

The full study guide: modules, problems, answer key
Module 1 · Grades, Slopes & Earthwork
Applied Field Engineering-1 Percent Grade from Rise and Run
G=100 ΔhLG = \frac{100\,\Delta h}{L}
Percent grade is one hundred times the vertical rise divided by the horizontal run, measured level,…
Applied Field Engineering-2 Elevation from Grade and Distance
E2=E1+G L100E_2 = E_1 + \frac{G\,L}{100}
Projects an elevation along a uniform grade, adding the rise over a measured horizontal distance to…
Applied Field Engineering-3 Grade to Slope Angle
θ=arctan⁡ ⁣(G100)\theta = \arctan\!\left(\frac{G}{100}\right)
Converts a percent grade into the slope angle measured from horizontal, and back again with the…
Applied Field Engineering-4 Slope Ratio (H:V) to Percent Grade
G=100nG = \frac{100}{n}
Converts an embankment slope quoted as n horizontal to one vertical into the equivalent percent…
Applied Field Engineering-5 Earthwork Volume by Average End Area
V=L (A1+A2)2V = \frac{L\,(A_1 + A_2)}{2}
Volume between two cross sections, averaging their end areas over the distance between them; the…
Applied Field Engineering-6 Earthwork Volume by the Prismoidal Formula
V=L (A1+4Am+A2)6V = \frac{L\,(A_1 + 4A_m + A_2)}{6}
Simpson's rule applied to earthwork, weighting the middle cross section four times the ends; the…
Applied Field Engineering-7 Swell: Loose Volume from Bank Volume
VL=VB(1+S100)V_L = V_B\left(1 + \frac{S}{100}\right)
Converts undisturbed bank volume into the loose volume the same soil occupies once excavated, using…
Applied Field Engineering-8 Shrinkage: Compacted Volume from Bank Volume
VC=VB(1−Sh100)V_C = V_B\left(1 - \frac{S_h}{100}\right)
Converts bank volume into the compacted volume it fills in an engineered embankment, using the…
Applied Field Engineering-9 Concrete Volume with Waste Allowance
V=L W T(1+w100)V = L\,W\,T\left(1 + \frac{w}{100}\right)
Orders concrete for a rectangular slab, footing or column by adding a percent waste allowance to…
Applied Field Engineering-10 Asphalt Tonnage from Area and Thickness
M=A t ρM = A\,t\,\rho
Converts a paving area, lift thickness and compacted mix density into the tonnage to order,…
Applied Field Engineering-11 Stockpile Volume (Truncated Pyramid)
V=h3(A1+A2+A1A2)V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)
Volume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer…
Module 2 · Curves, Traverse & Levelling
Applied Field Engineering-12 Radius from Degree of Curve (Arc Definition)
R=5729.578DR = \frac{5729.578}{D}
Converts degree of curve to radius using the arc definition, where D is the central angle…
Applied Field Engineering-13 Horizontal Curve Tangent Length
T=Rtan⁡Δ2T = R\tan\frac{\Delta}{2}
Distance from the point of intersection back to the point of curvature, from the curve radius and…
Applied Field Engineering-14 Horizontal Curve Length from Degree of Curve
L=100 ΔDL = \frac{100\,\Delta}{D}
Length of a circular curve in 100 ft stations, from the total deflection angle and the degree of…
Applied Field Engineering-15 Horizontal Curve Long Chord
C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}
Straight-line distance from the point of curvature to the point of tangency, the chord that spans…
Applied Field Engineering-16 Horizontal Curve External Distance
E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)
Clearance from the point of intersection to the midpoint of the curve, the distance a curve cuts…
Applied Field Engineering-17 Horizontal Curve Middle Ordinate
M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)
Offset from the middle of the long chord to the middle of the arc, the number that governs sight…
Applied Field Engineering-18 Deflection Angle to a Point on a Circular Curve
δ=ℓ2R\delta = \frac{\ell}{2R}
Angle turned off the back tangent at the PC to sight a point a given arc distance along a circular…
Applied Field Engineering-19 Vertical Curve Length from K Value
L=K AL = K\,A
Length of a parabolic vertical curve as the design K value times the algebraic grade change, K…
Applied Field Engineering-20 Elevation on a Parabolic Vertical Curve
E=EBVC+g1x100+A x2200 LE = E_{BVC} + \frac{g_1 x}{100} + \frac{A\,x^{2}}{200\,L}
Elevation at any station on an equal-tangent parabolic vertical curve, measured from the beginning…
Applied Field Engineering-21 High or Low Point on a Vertical Curve
xt=−g1LAx_t = -\frac{g_1 L}{A}
Distance from the BVC to the crest or sag of a parabolic vertical curve, where the profile grade…
Applied Field Engineering-22 Stopping Sight Distance
d=v tr+v22ad = v\,t_r + \frac{v^{2}}{2a}
Distance to stop from a design speed, adding the reaction distance travelled before braking to the…
Applied Field Engineering-23 Crest Vertical Curve Length for Sight Distance
L=A S2200(h1+h2)2L = \frac{A\,S^{2}}{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}
Crest curve length needed to see an object over the hill, for the case where sight distance is…
Applied Field Engineering-24 Superelevation Rate for a Horizontal Curve
e100+f=v2g R\frac{e}{100} + f = \frac{v^{2}}{g\,R}
Balances banking and side friction against centripetal demand on a horizontal curve, using standard…
Applied Field Engineering-25 Back Azimuth
αb=α±180∘\alpha_b = \alpha \pm 180^{\circ}
Reverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside…
Applied Field Engineering-26 Latitude of a Traverse Leg
Lat=Lcos⁡α\text{Lat} = L\cos\alpha
North-south component of a traverse course, from the measured slope-corrected distance and the…
Applied Field Engineering-27 Departure of a Traverse Leg
Dep=Lsin⁡α\text{Dep} = L\sin\alpha
East-west component of a traverse course, taken from the measured horizontal distance and the…
Applied Field Engineering-28 Traverse Closure Error
Ec=(ΣLat)2+(ΣDep)2E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}
Linear misclosure of a closed traverse, combining the residual sums of the latitudes and departures…
Applied Field Engineering-29 Traverse Precision Ratio
N=PEcN = \frac{P}{E_c}
Expresses traverse accuracy as one part in N, dividing the total perimeter walked by the linear…
Applied Field Engineering-30 Differential Levelling Elevation
E2=E1+BS−FSE_2 = E_1 + \text{BS} - \text{FS}
Carries an elevation forward one instrument setup, adding the backsight to the known point and…
Applied Field Engineering-31 Stadia Distance from Rod Intercept
D=K s+CD = K\,s + C
Horizontal distance from a stadia rod intercept, with the instrument's stadia interval factor and…
Applied Field Engineering-32 Area of a Three-Sided Parcel by Coordinates
A=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]
Shoelace area of a parcel from the coordinates of its three corners, listed counter-clockwise so…
Module 3 · Stacks & Plumes
Applied Field Engineering-33 Environmental Lapse Rate
Γ=T1−T2z2−z1\Gamma = \frac{T_1 - T_2}{z_2 - z_1}
How fast the air cools with height, from two temperatures at two altitudes.
Applied Field Engineering-34 ppm to mg/m³ Conversion
C=ppm⋅M24.45C = \frac{ppm \cdot M}{24.45}
Converts a gas concentration by volume into one by mass, using the molar volume of an ideal gas at…
Applied Field Engineering-35 Barometric Pressure with Altitude
P=P0 e−Mgz/RTP = P_0 \, e^{-Mgz/RT}
Atmospheric pressure at a height above a reference level, for an isothermal layer.
Applied Field Engineering-36 Wind Speed at Height (Power Law)
u2=u1(z2z1)pu_2 = u_1 \left(\frac{z_2}{z_1}\right)^{p}
Extrapolates a wind measurement to another height using the power-law profile.
Applied Field Engineering-37 Stack Exit Velocity
vs=4Qvπd2v_s = \frac{4 Q_v}{\pi d^2}
The speed exhaust leaves a round stack, from the volumetric flow and the inside diameter.
Applied Field Engineering-38 Stack Draft Pressure (Chimney Effect)
Δp=h g (ρa−ρs)\Delta p = h \, g \, (\rho_a - \rho_s)
The pressure a chimney generates on its own, from the height of the column and the density…
Applied Field Engineering-39 Good Engineering Practice Stack Height
HGEP=hb+1.5LH_{GEP} = h_b + 1.5 L
The stack height that just clears the turbulent wake of a nearby building, by the US EPA formula:…
Applied Field Engineering-40 Briggs Buoyancy Flux
F=g vs d2(Ts−Ta)4 TsF = \frac{g \, v_s \, d^2 (T_s - T_a)}{4 \, T_s}
The buoyancy flux parameter of a hot stack plume, from exit velocity, stack diameter and the…
Applied Field Engineering-41 Briggs Plume Rise (Neutral and Unstable)
Δh=1.6 F1/3x2/3u\Delta h = \frac{1.6 \, F^{1/3} x^{2/3}}{u}
How far a buoyant plume climbs above the stack at a given downwind distance, in neutral or unstable…
Applied Field Engineering-42 Holland Plume Rise
Δh=vsdu(1.5+2.68×10−3Pd Ts−TaTs)\Delta h = \frac{v_s d}{u}\left(1.5 + 2.68\times10^{-3} P d \, \frac{T_s - T_a}{T_s}\right)
Plume rise by Holland's 1953 equation, combining momentum and buoyancy in one step.
Applied Field Engineering-43 Effective Stack Height
H=hs+ΔhH = h_s + \Delta h
The height dispersion actually starts from: the physical stack plus the plume rise.
Applied Field Engineering-44 Emission Rate from Stack Concentration
E=C QvE = C \, Q_v
The mass of pollutant leaving a stack per unit time, from the measured concentration and the…
Applied Field Engineering-45 Emission Correction to Reference Oxygen
Ccorr=Cmeas 20.9−O2,ref20.9−O2,measC_{corr} = C_{meas} \, \frac{20.9 - O_{2,ref}}{20.9 - O_{2,meas}}
Restates a measured stack concentration at the reference oxygen a limit is written at, so that…
Applied Field Engineering-46 Excess Air from Flue Gas Oxygen
EA=O220.9−O2EA = \frac{O_2}{20.9 - O_2}
How much air beyond stoichiometric is passing through a burner, read straight off the oxygen in the…
Applied Field Engineering-47 Pasquill–Gifford Dispersion Coefficient
σ=a xb\sigma = a \, x^{b}
The crosswind or vertical spread of a plume at a downwind distance, in the power-law form fitted to…
Applied Field Engineering-48 Gaussian Plume Ground-Level Concentration
C=Qπσyσzu e−H2/(2σz2)C = \frac{Q}{\pi \sigma_y \sigma_z u} \, e^{-H^{2}/(2\sigma_z^{2})}
Ground-level concentration directly downwind of an elevated point source, on the plume centreline,…
Applied Field Engineering-49 Maximum Ground-Level Concentration
Cmax=2QeπuH2⋅σzσyC_{max} = \frac{2Q}{e \pi u H^{2}} \cdot \frac{\sigma_z}{\sigma_y}
The worst ground-level concentration a plume ever produces, wherever downwind it occurs.
Applied Field Engineering-50 Particulate Collection Efficiency
η=Cin−CoutCin\eta = \frac{C_{in} - C_{out}}{C_{in}}
The fraction of the dust entering a control device that does not leave it, from inlet and outlet…
Applied Field Engineering-51 Isokinetic Sampling Rate
Qn=vsAnQ_n = v_s A_n
The flow a sampling train must draw so that gas enters the nozzle at exactly the stack velocity.
Module 4 · Noise on Site
Applied Field Engineering-52 Sound Power Level to Sound Pressure Level
Lp=LW+10log⁡10 ⁣(Q4πr2)L_p = L_W + 10\log_{10}\!\left(\frac{Q}{4\pi r^{2}}\right)
A machine's sound POWER is a property of the machine; its sound PRESSURE is what a meter reads at a…
Applied Field Engineering-53 Combining Sound Levels
Lt=10log⁡10 ⁣(10L1/10+10L2/10)L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)
Decibels do not add.
Applied Field Engineering-54 Distance Attenuation from a Point Source
L2=L1−20log⁡10 ⁣(r2r1)L_2 = L_1 - 20\log_{10}\!\left(\frac{r_2}{r_1}\right)
Move twice as far from a point source in the open and the level drops 6 dB: every time, regardless…
Applied Field Engineering-55 Sabine Reverberation Time (RT60)
T60=0.161 VAT_{60} = \frac{0.161\,V}{A}
How long a sound takes to fade by 60 decibels after the source stops: the single number that…
Applied Field Engineering-56 Total Absorption (Sabins)
A=S1α1+S2α2+S3α3A = S_1\alpha_1 + S_2\alpha_2 + S_3\alpha_3
Add up the room: every surface contributes its area multiplied by how much of the sound striking it…
Applied Field Engineering-57 Eyring Reverberation Time
T60=0.161 V−S ln⁡(1−αˉ)T_{60} = \frac{0.161\,V}{-S\,\ln(1-\bar{\alpha})}
Sabine's equation, corrected for rooms that actually absorb.
Applied Field Engineering-58 Noise Reduction Coefficient (NRC)
NRC=α250+α500+α1000+α20004\mathrm{NRC} = \frac{\alpha_{250} + \alpha_{500} + \alpha_{1000} + \alpha_{2000}}{4}
The single number on an acoustic panel's data sheet: the plain average of its absorption…
Applied Field Engineering-59 Sound Transmission Loss
TL=10log⁡10 ⁣(IiIt)TL = 10\log_{10}\!\left(\frac{I_i}{I_t}\right)
How much of the sound striking a partition never gets out the other side, in decibels.
Applied Field Engineering-60 Mass Law Transmission Loss
TL=20log⁡10 ⁣(πmfρ0c)−5TL = 20\log_{10}\!\left(\frac{\pi m f}{\rho_0 c}\right) - 5
For a single limp panel, blocking sound is almost entirely about weight: transmission loss climbs…
Applied Field Engineering-61 Composite Transmission Loss
TLc=10log⁡10 ⁣(Sw+SdSw 10−TLw/10+Sd 10−TLd/10)TL_c = 10\log_{10}\!\left(\frac{S_w + S_d}{S_w\,10^{-TL_w/10} + S_d\,10^{-TL_d/10}}\right)
A wall with a door or a window in it, rated as one partition.
Applied Field Engineering-62 Allowable Noise Exposure Time
T=82(L−Lc)/qT = \frac{8}{2^{(L - L_c)/q}}
How long a person may be exposed to a steady noise level before reaching a full daily dose.
Applied Field Engineering-63 Day–Night Average Sound Level (Ldn)
Ldn=10log⁡10 ⁣(15⋅10Ld/10+9⋅10(Ln+10)/1024)L_{dn} = 10\log_{10}\!\left(\frac{15 \cdot 10^{L_d/10} + 9 \cdot 10^{(L_n + 10)/10}}{24}\right)
The twenty-four-hour energy average of a community's noise with a 10 dB penalty added to every…
Module 5 · Water on the Land
Applied Field Engineering-64 Available Water Capacity
AWC=(θfc−θpwp) DAWC = (\theta_{fc} - \theta_{pwp}) \, D
The depth of water a soil can hold that a crop can actually use: the gap between field capacity and…
Applied Field Engineering-65 Readily Available Water from MAD
RAW=AWC×MADRAW = AWC \times MAD
The depth of water a manager is willing to let the crop draw down before irrigating: the available…
Applied Field Engineering-66 Crop Evapotranspiration
ETc=Kc×ET0ET_c = K_c \times ET_0
The water a specific crop uses on a given day, from a reference evapotranspiration that describes…
Applied Field Engineering-67 Irrigation Interval
I=RAWETcI = \frac{RAW}{ET_c}
How many days a crop can go between irrigations: the depth of water it is allowed to use, divided…
Applied Field Engineering-68 Net Irrigation Requirement
IRn=ETc−PeIR_n = ET_c - P_e
The depth of water irrigation must supply over a period: everything the crop used, less the…
Applied Field Engineering-69 Irrigation Application Efficiency
Ea=WsWdE_a = \frac{W_s}{W_d}
The share of the water delivered to a field that ends up stored in the root zone where the crop can…
Applied Field Engineering-70 Distribution Uniformity
DU=dˉlqdˉDU = \frac{\bar{d}_{lq}}{\bar{d}}
How evenly an irrigation system puts water on the ground: the average depth caught in the driest…
Applied Field Engineering-71 Irrigation Set Run Time
t=d AQt = \frac{d \, A}{Q}
How long a set must run to lay down a wanted depth of water over a known area at a known flow rate:…
Applied Field Engineering-72 Irrigation System Capacity
Q=A ETpEa fQ = \frac{A \, ET_p}{E_a \, f}
The flow an irrigation system must deliver to keep up with peak crop water use, once application…
Applied Field Engineering-73 Sprayer Application Rate
V=Qw vV = \frac{Q}{w \, v}
Spray volume applied per unit of land from one nozzle's output, the nozzle spacing and the ground…
Applied Field Engineering-74 Nozzle Output at a New Pressure
Q2=Q1p2p1Q_2 = Q_1 \sqrt{\frac{p_2}{p_1}}
How a spray nozzle's output changes when the pressure changes: flow rises with the square root of…
Applied Field Engineering-75 Effect of Speed on Application Rate
V2=V1v1v2V_2 = V_1 \frac{v_1}{v_2}
How the applied rate changes when ground speed changes and nothing else does: rate and speed are…
Applied Field Engineering-76 Nozzle Output Deviation
D=100 Qm−QrQrD = 100 \, \frac{Q_m - Q_r}{Q_r}
How far a nozzle has drifted from its rated output, as a percentage.
Applied Field Engineering-77 Area Covered per Tank
A=TVA = \frac{T}{V}
How much ground one tankful covers: the tank volume divided by the application rate.
Applied Field Engineering-78 Tank Loads to Cover a Field
N=A VTN = \frac{A \, V}{T}
How many tankfuls a field will take: the area times the application rate, divided by the tank…
Applied Field Engineering-79 Active Ingredient Rate
Rai=c RvR_{ai} = c \, R_v
The active ingredient actually applied per unit area, from the formulation's concentration and the…
Applied Field Engineering-80 Percent Solution in the Tank
P=100 VpTP = 100 \, \frac{V_p}{T}
The strength of a spray mix stated as a percentage of the tank: product volume over tank volume.
Applied Field Engineering-81 Seeding Rate from Target Plant Population
S=P⋅TKW1000 G ES = \frac{P \cdot TKW}{1000 \, G \, E}
The seeding rate in weight per unit area that establishes a target plant population, from the…
Applied Field Engineering-82 Plant Population from Row and Seed Spacing
P=1w sP = \frac{1}{w \, s}
The plant population a geometry produces: one plant occupies a rectangle of row spacing by in-row…
Applied Field Engineering-83 Pure Live Seed
PLS=p×gPLS = p \times g
The fraction of a seed lot that is both the species you paid for and capable of growing.
Applied Field Engineering-84 Field Emergence
E=PSE = \frac{P}{S}
The share of sown seed that became a counted plant.
Applied Field Engineering-85 Effective Field Capacity
C=w S eC = w \, S \, e
How fast a machine actually covers ground, from its working width, its field speed and its field…
Applied Field Engineering-86 Field Efficiency
e=CeCte = \frac{C_e}{C_t}
The share of a machine's theoretical capacity it actually delivers, once turning, filling,…
Applied Field Engineering-87 Time to Cover a Field
t=ACt = \frac{A}{C}
The hours a field takes at a given effective field capacity.
Module 6 · Navigation & Field Craft
Applied Field Engineering-88 Map Scale to Real Distance
d=m Sd = m\,S
Real distance from a map measurement times the scale denominator.
Applied Field Engineering-89 Distance from Pace Count
d=n Ld = n\,L
Distance covered from a pace count times your calibrated pace length.
Applied Field Engineering-90 Ground Distance on a Slope
g=m2+r2g = \sqrt{m^{2} + r^{2}}
The distance your legs travel on a slope, from map distance and rise.
Applied Field Engineering-91 Height by Clinometer
H=dtan⁡θ+eH = d\tan\theta + e
Height of a tree or cliff from a paced distance and a clinometer angle, plus your eye height: the…
Applied Field Engineering-92 Naismith's Rule (Hiking Time)
t=d5 km/h+h600 m/ht = \frac{d}{5\,\text{km/h}} + \frac{h}{600\,\text{m/h}}
Naismith's 1892 walking-time estimate: an hour for every 5 km on the map plus an hour for every 600…
Applied Field Engineering-93 Estimated Time En Route
t=dVgt = \frac{d}{V_g}
Distance divided by ground speed.
Applied Field Engineering-94 Compass to True Heading (Variation and Deviation)
T=C+D+VT = C + D + V
The TVMDC chain in one line: a compass heading corrected by deviation for the ship's own iron and…
Applied Field Engineering-95 Great Circle Distance (Haversine)
d=2Rarcsin⁡sin⁡2φ2−φ12+cos⁡φ1cos⁡φ2sin⁡2λ2−λ12d = 2R\arcsin\sqrt{\sin^{2}\frac{\varphi_2-\varphi_1}{2} + \cos\varphi_1\cos\varphi_2\sin^{2}\frac{\lambda_2-\lambda_1}{2}}
The shortest distance between two points on a sphere, from their latitudes and longitudes.
Applied Field Engineering-96 Initial Great Circle Bearing
θ=atan2⁡ ⁣(sin⁡Δλ cos⁡φ2,  cos⁡φ1sin⁡φ2−sin⁡φ1cos⁡φ2cos⁡Δλ)\theta = \operatorname{atan2}\!\left(\sin\Delta\lambda\,\cos\varphi_2,\; \cos\varphi_1\sin\varphi_2 - \sin\varphi_1\cos\varphi_2\cos\Delta\lambda\right)
The course at the moment of departure along the great circle to a destination, measured clockwise…
Applied Field Engineering-97 Rhumb Line Distance
d=R(φ2−φ1)2+q2(λ2−λ1)2d = R\sqrt{\left(\varphi_2-\varphi_1\right)^{2} + q^{2}\left(\lambda_2-\lambda_1\right)^{2}}
The length of the track that crosses every meridian at the same angle: the route you actually steer…
Applied Field Engineering-98 Dead Reckoning Position
Δφ=S tcos⁡CR\Delta\varphi = \frac{S\,t\cos C}{R}
The classic DR advance: speed times time gives the distance run, and the cosine of the course…
Applied Field Engineering-99 Set and Drift of the Current
Dr=Δn2+Δe2tD_r = \frac{\sqrt{\Delta n^{2} + \Delta e^{2}}}{t}
The current the water was making, worked backwards from the gap between where dead reckoning said…
Applied Field Engineering-100 Cross Track Error
ext=Rarcsin⁡(sin⁡d13R sin⁡(θ13−θ12))e_{xt} = R\arcsin\left(\sin\frac{d_{13}}{R}\,\sin\left(\theta_{13}-\theta_{12}\right)\right)
The perpendicular distance from the intended track to where you actually are, on a sphere: the…
Applied Field Engineering-101 Wind Triangle Ground Speed
Vg=V2−W2sin⁡2θ−Wcos⁡θV_g = \sqrt{V^{2} - W^{2}\sin^{2}\theta} - W\cos\theta
What the aircraft is actually making over the ground once the wind has had its say: true airspeed,…
Applied Field Engineering-102 Wind Correction Angle
WCA=arcsin⁡(Wsin⁡θV)\mathrm{WCA} = \arcsin\left(\frac{W\sin\theta}{V}\right)
How far off the desired track to point the nose so the wind carries you along it instead of off it.
Applied Field Engineering-103 Distance to the Visible Horizon
D=h(2R+h)D = \sqrt{h\left(2R + h\right)}
How far away the sea horizon is from a given height of eye: the tangent line from the observer to…
Applied Field Engineering-104 Longitude to Solar Time Difference
Δt=4 Δλ\Delta t = 4\,\Delta\lambda
Solar time difference from a longitude difference: 4 minutes per degree, because the Earth turns…
Module 7 · Engineering Economics
Applied Field Engineering-105 Present Value
PV=FV(1+r)t\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}
Today's value of a future amount, discounted at rate r per period.
Applied Field Engineering-106 Compound Interest (Periodic)
A=P(1+rn)ntA = P \left( 1 + \frac{r}{n} \right)^{n t}
Amount after t periods when interest compounds n times per period at rate r.
Applied Field Engineering-107 Effective Annual Rate from a Nominal Rate
EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1
What a quoted nominal annual rate really costs or earns once it compounds m times a year: 12%…
Applied Field Engineering-108 Real Interest Rate (Fisher Equation)
rreal=1+i1+f−1r_{\text{real}} = \frac{1 + i}{1 + f} - 1
What a return is worth after inflation is taken out, done exactly rather than by the rough…
Applied Field Engineering-109 Rule of 72 (Doubling Time)
n≈0.72in \approx \frac{0.72}{i}
Mental-arithmetic estimate of how many periods it takes money to double at rate i: with the rate…
Applied Field Engineering-110 Future Value of an Annuity (Regular Deposits)
FV=D (1+i)n−1i\mathit{FV} = D\,\frac{(1+i)^n - 1}{i}
What an equal deposit made at the end of every period grows to after n periods at rate i: the…
Applied Field Engineering-111 Net Present Value of a Uniform Annual Cash Flow
NPV=A 1−(1+i)−ni−C0\mathit{NPV} = A\,\frac{1 - (1+i)^{-n}}{i} - C_0
Worth of a project today when it costs C0 up front and returns the same net amount every year for n…
Applied Field Engineering-112 Capital Recovery Factor
CRF=i (1+i)n(1+i)n−1\mathit{CRF} = \frac{i\,(1+i)^{n}}{(1+i)^{n} - 1}
Fraction of a capital sum that must be recovered each year to repay it with interest over n years.
Applied Field Engineering-113 Loan Payment (Amortized Loan or Mortgage)
M=P i1−(1+i)−nM = \frac{P\,i}{1 - (1+i)^{-n}}
Level payment that retires a loan of principal P in exactly n payments at periodic interest rate i:…
Applied Field Engineering-114 Total Interest Paid Over a Loan
I=M n−PI = M\,n - P
Everything a loan costs beyond the amount borrowed: total of all payments minus the principal.
Applied Field Engineering-115 Sinking Fund Factor (A/F)
(A/F,i,n)=i(1+i)n−1(A/F, i, n) = \frac{i}{(1+i)^{n} - 1}
The engineering-economy factor (A/F, i, n): the fraction of a future amount that must be set aside…
Applied Field Engineering-116 Equivalent Annual Cost
EAC=P i (1+i)n(1+i)n−1+M\mathit{EAC} = P\,\frac{i\,(1+i)^{n}}{(1+i)^{n} - 1} + M
Total yearly cost of owning and running an asset: the capital cost spread over its life at…
Applied Field Engineering-117 Benefit–Cost Ratio
B/C=BCB/C = \frac{B}{C}
Present worth of a project's benefits divided by the present worth of its costs.
Applied Field Engineering-118 Simple Payback Period
t=CSt = \frac{C}{S}
Years for an upgrade to repay its own capital cost out of the money it saves each year, ignoring…
Applied Field Engineering-119 Return on Investment (ROI)
ROI=G−CC\mathit{ROI} = \frac{G - C}{C}
Profit expressed as a fraction of what was spent: total value returned minus cost, divided by cost.
Applied Field Engineering-120 Break-Even Quantity
Q=Fp−vQ = \frac{F}{p - v}
Units that must be sold before fixed costs are covered: fixed cost divided by the contribution…
Applied Field Engineering-121 Straight-Line Depreciation
D=C−SnD = \frac{C - S}{n}
Equal annual write-down of an asset: cost minus salvage value, spread evenly over its useful life.
Applied Field Engineering-122 Declining-Balance Depreciation (Book Value)
B=C (1−d)kB = C\,(1 - d)^k
Book value of an asset after k years when a fixed fraction d of the remaining value is written off…