Calculations

Calculations

• Half-width of a contour

  Half the crosswind width of a concentration level at a given distance.

  w = σy × √(2 × ln(C₀ ÷ C_end))

  Gaussian plume concentration and contours process/env/plume.js

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• Height of a contour

  The height above ground up to which the concentration stays at or above the endpoint at a given distance.

  Gaussian plume concentration and contours process/env/plume.js

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• Contour polygon in site coordinates

  The outline of a concentration level as a polygon on the site plan, with its reach and greatest height.

  (x, y) → (cx + x·cos t − y·sin t, cy + x·sin t + y·cos t)

  Gaussian plume concentration and contours process/env/plume.js

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• Gas density relative to air

  Whether the released gas is heavier than air, which makes a Gaussian result untrustworthy.

  ratio = (P·M ÷ R·T) ÷ (P·M_air ÷ R·T_air); dense when ratio > 1.1

  Gaussian plume concentration and contours process/env/plume.js

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• Wind speed at height

  The wind speed at a height from the speed measured at 10 m.

  u = u₁₀ × (max(z, 1) ÷ 10)^p

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Plume rise and effective height

  How far the plume rises above the stack and the effective release height used by the plume equation.

  Fb = g·vs·d²·(Ts − Ta) ÷ 4Ts; Δh = 21.425·Fb^0.75 ÷ u (Fb < 55) or 38.71·Fb^0.6 ÷ u; stable: Δh = 2.6·∛(Fb ÷ u·s)

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Stack from a flowsheet stream

  Exit velocity, volume flow and the emission rate of each species from a process stream.

  V̇ = F·R·T ÷ P; vs = V̇ ÷ (π·d² ÷ 4); Qᵢ = F·zᵢ·MWᵢ

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Concentration from one stack at one point

  The ground concentration in µg/m³ from one source for one wind speed and stability class.

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Worst-case 1-hour concentration

  The highest ground-level concentration over all stability classes, wind speeds and distances, and its longer-period estimates.

  xᵢ = xMin × (xMax ÷ xMin)^(i ÷ steps); C_avg = C_max × factor

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Receptor position in the wind frame

  The downwind and crosswind distance of a receptor from a source for one hour's wind direction, and the concentration there.

  θ = (dirFrom + 180) mod 360; x_d = dx·sin θ + dy·cos θ; y_c = dx·cos θ − dy·sin θ

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Hour-by-hour statistics at receptors

  The highest and second-highest hourly value, the period mean and the highest 24-hour mean at each receptor.

  mean = Σ cₕ ÷ n; rolling₂₄ = max over windows of (Σ of 24 hours) ÷ 24

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Contours for a site plan

  The contour polygon of each concentration level for one wind condition, in site coordinates.

  angle = 90 − ((dirFrom + 180) mod 360); C_end = level × 10⁻⁶

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Comparison with a limits pack

  Whether a result is above, or within 80 % of, the limit a person typed for that pollutant and averaging time.

  fraction = c ÷ limit

  Stack plume rise and air dispersion screening process/env/dispersion.js

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• Contours as drawing entities

  The contour polylines, stack marker, wind arrow and labels placed in a drawing.

  drawing point = site point × scale

  Stack plume rise and air dispersion screening process/env/plumedraw.js

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• Stoichiometric matrix

  The 8 by 13 table of how much of each component each process makes or uses.

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Process rates

  The rate of each of the 8 biological processes at given concentrations.

  r₁ = µH × SS ÷ (KS + SS) × SO ÷ (KOH + SO) × XBH (and seven more Monod products)

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Reaction rate of each component

  The net production of each of the 13 components from the 8 process rates.

  dcₖ/dt = Σⱼ νⱼₖ × ρⱼ

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Clarifier split

  The effluent and underflow concentrations from the clarifier feed and the capture fraction a person gives.

  eff = (1 − capture) × c; und = (Qc × c − Qe × eff) ÷ Qu

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Tank mass balances

  The rate of change of every component in every tank: flow in and out, reaction and aeration.

  dc/dt = (Σ Q_in × c_in − Q × c) ÷ V + Σⱼ νⱼ × ρⱼ + KLa × (SO_sat − SO)

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Steady-state iteration

  Advances the plant state with implicit steps until nothing changes any more.

  (I ÷ h − J) × Δy = F(y); stop when max |Fᵢ| ÷ (|yᵢ| + 1) < tol

  Activated sludge plant steady state (ASM1) process/env/water.js

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• COD, nitrogen and oxygen balances, sludge age

  Checks that what enters the plant equals what leaves plus what reacts, and gives the sludge retention time.

  residual = in − out + reaction; rel = residual ÷ (|in| + 10⁻¹²); SRT = Σ V × solids ÷ (Qw × Xw + Qe × Xe)

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Oxygen used and supplied

  The oxygen the reactions use next to what the aerators supply, in kg per day.

  kg/d = g/d ÷ 1000

  Activated sludge plant steady state (ASM1) process/env/water.js

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• Bar-screen head loss

  The head lost through a bar screen.

  h = β × (w ÷ b)^(4/3) × v² ÷ 2g × sin θ

  Water treatment unit processes and sludge sizing process/env/water.js

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• Ideal settling basin

  The overflow rate and the fraction of particles removed.

  v₀ = Q ÷ A; removal = min(1, vs ÷ v₀)

  Water treatment unit processes and sludge sizing process/env/water.js

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• Granular filter head loss

  The head lost through a clean filter bed in laminar flow.

  h = 180 × L × ν × (1 − ε)² × v ÷ (g × ε³ × d²)

  Water treatment unit processes and sludge sizing process/env/water.js

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• Disinfection survival

  The fraction of organisms surviving a contact time, and the log removal.

  s = exp(−k × Cⁿ × t); log removal = −log₁₀(s)

  Water treatment unit processes and sludge sizing process/env/water.js

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• Contact time for a log removal

  The contact time needed to reach a wanted log removal.

  t = log₁₀ removal × ln 10 ÷ (k × Cⁿ)

  Water treatment unit processes and sludge sizing process/env/water.js

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• Sludge retention time

  The mass of sludge in the system divided by the mass leaving per day.

  SRT = V × X ÷ (Qw × Xw + Qe × Xe)

  Water treatment unit processes and sludge sizing process/env/water.js

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• Tank footprints for the site plan

  The plan area and the side of a square footprint for each tank.

  area = V ÷ depth; side = √area

  Water treatment unit processes and sludge sizing process/env/water.js

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• Complete-mix tank sizing

  Sludge production, tank volume, retention time, food-to-mass ratio and oxygen demand.

  Px = Y × Q × (S₀ − S) ÷ (1 + kd × θc); V = θc × Px ÷ X; O₂ = Q × (S₀ − S) − 1.42 × Px

  Water treatment unit processes and sludge sizing process/env/design.js

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• Return sludge ratio

  The return flow as a fraction of the plant flow.

  Qr ÷ Q = (X − Xe) ÷ (Xr − X)

  Water treatment unit processes and sludge sizing process/env/design.js

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• Clarifier loadings

  Surface overflow rate and solids loading of a clarifier.

  overflow = Q ÷ A; solids = (Q + Qr) × X ÷ 1000 ÷ A ÷ 24

  Water treatment unit processes and sludge sizing process/env/design.js

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• Reading to SI units

  Converts a temperature, pressure, flow or length reading to SI, including gauge pressures.

  K = °C + 273.15; K = (°F − 32) × 5 ÷ 9 + 273.15; Pa = gauge × factor + atmosphere; else value × factor

  Plant readings: units, comparison and trends process/env/historian.js

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• Marker text

  The value shown next to a tag on the diagram and the 3D model.

  Plant readings: units, comparison and trends process/env/historian.js

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• Plant against model

  How far each plant reading is from the value the model gives.

  deviation = (plant − model) ÷ |model| (÷ 1 when the model value is 0)

  Plant readings: units, comparison and trends process/env/historian.js

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• Trend thinning

  Reduces a long series to at most a given number of points for plotting, keeping its shape.

  per bucket: keep the sample with the largest triangle area against the previous kept point and the next bucket's mean

  Plant readings: units, comparison and trends process/env/historian.js

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• Series summary

  Count, minimum, maximum, mean and standard deviation of a series.

  mean = Σ v ÷ n; sd = √(Σ (v − mean)² ÷ n)

  Plant readings: units, comparison and trends process/env/historian.js

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• Log-mean temperature difference

  The mean driving temperature difference of a counter-current exchanger.

  ΔT_lm = (ΔT₁ − ΔT₂) ÷ ln(ΔT₁ ÷ ΔT₂)

  Heat exchanger rating process/equip/exchanger.js

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• LMTD correction factor

  The correction for one shell pass with an even number of tube passes.

  F = [S ÷ (R − 1)] × ln((1 − P) ÷ (1 − R·P)) ÷ ln((2 − P(R + 1 − S)) ÷ (2 − P(R + 1 + S))), S = √(R² + 1)

  Heat exchanger rating process/equip/exchanger.js

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• Effectiveness from NTU

  The fraction of the largest possible heat transfer an exchanger of a given type achieves.

  counter: ε = (1 − e^(−NTU(1 − Cr))) ÷ (1 − Cr·e^(−NTU(1 − Cr))); Cr → 0: ε = 1 − e^(−NTU)

  Heat exchanger rating process/equip/exchanger.js

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• NTU for a wanted effectiveness

  The size (NTU) an exchanger needs to reach a given effectiveness.

  Heat exchanger rating process/equip/exchanger.js

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• Duty and outlet temperatures from UA

  The heat duty and both outlet temperatures of an exchanger of known UA.

  Q = ε × C_min × (T_h,in − T_c,in); T_h,out = T_h,in − Q ÷ C_h; T_c,out = T_c,in + Q ÷ C_c

  Heat exchanger rating process/equip/exchanger.js

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• Tube-side film coefficient

  The heat-transfer coefficient and friction factor inside the tubes.

  f = (0.79·ln Re − 1.64)⁻²; Nu = (f/8)(Re − 1000)·Pr ÷ (1 + 12.7·√(f/8)·(Pr^(2/3) − 1))

  Heat exchanger rating process/equip/exchanger.js

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• Overall heat-transfer coefficient

  The overall coefficient referred to the outside tube area, from both films, the wall and fouling.

  1 ÷ U = 1 ÷ h_o + R_fo + d_o·ln(d_o ÷ d_i) ÷ 2k_w + R_fi·d_o ÷ d_i + d_o ÷ (d_i·h_i)

  Heat exchanger rating process/equip/exchanger.js

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• Shell-side coefficient and pressure drop (Kern)

  The film coefficient and pressure drop on the shell side by Kern's screening method.

  h = 0.36·(k ÷ De)·Re^0.55·Pr^(1/3)·φ; ΔP = f·Gs²·Ds·(Nb + 1) ÷ (2ρ·De·φ); f = exp(0.576 − 0.19·ln Re)

  Heat exchanger rating process/equip/exchanger.js

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• Tube layout and count

  The position of every tube inside the shell and how many fit.

  x = i × pitch (+ pitch ÷ 2 on odd triangular rows); y = j × pitch × √3 ÷ 2; kept when x² + y² ≤ R²

  Heat exchanger rating process/equip/exchanger.js

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• Shell-and-tube rating

  Duty, mean temperature difference, overall coefficient, required and provided area, overdesign and pressure drops.

  A_req = Q ÷ (U × F × ΔT_lm); A_prov = n × π × d_o × L; overdesign = A_prov ÷ A_req − 1

  Heat exchanger rating process/equip/exchanger.js

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• Plate exchanger rating

  Film coefficients, overall coefficient, duty and outlet temperatures of a plate exchanger.

  Nu = C × Reⁿ × Pr^(1/3); U = 1 ÷ (1 ÷ h_h + 1 ÷ h_c + t ÷ k_w + R_f)

  Heat exchanger rating process/equip/exchanger.js

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• Tube vibration screening

  The natural frequency of a tube span, the vortex-shedding frequency and the critical gap velocity.

  f_n = (π ÷ 2) × √(E·I ÷ (m·L⁴)); I = π(d_o⁴ − d_i⁴) ÷ 64; V_c = K·f_n·d_o·√(2π·ζ·m ÷ (ρ·d_o²))

  Heat exchanger rating process/equip/exchanger.js

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• Baffled shell geometry

  Cross-flow area, leakage and bypass areas and the fraction of tubes in cross flow.

  Sm = B·[(Ds − Dotl) + (Dotl − d_o)(p − d_o) ÷ p]; Fc = [π + 2x·sin(acos x) − 2·acos x] ÷ π

  Heat exchanger rating process/equip/bell.js

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