Calculations

Calculations

• Complete combustion of a stream

  The CO2 and water produced, the oxygen needed and the heat released when a stream is burned.

  CO2 = Σ nᵢ × Cᵢ; H2O = Σ nᵢ × Hᵢ ÷ 2; O2 = Σ nᵢ × (Cᵢ + Hᵢ ÷ 4 − Oᵢ ÷ 2); kg CO2 = CO2 × 0.04401

  Process emissions (CO2 equivalent) process/emissions.js

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• Vented stream as CO2 equivalent

  The CO2-equivalent mass flow of a stream released to the air.

  CO2e = Σ F × zᵢ × MWᵢ × GWPᵢ

  Process emissions (CO2 equivalent) process/emissions.js

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• Emissions report

  The emission of each tagged stream, fired unit and electric unit, the total per second and the total per year.

  fired: CO2e = |Q| ÷ efficiency ÷ 1.055056e9 × factor; electric: CO2e = (|W| + |Q|) ÷ 3.6e6 × grid factor; per year = total × 3600 × hours

  Process emissions (CO2 equivalent) process/emissions.js

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• A stream's duty on one side of the pinch

  How much heat a stream has to give or take above the pinch, and below it.

  above, hot: Q = CP × (Ts − max(Tt, T_pinch)); above, cold: Q = CP × (Tt − max(Ts, T_pinch))

  Heat-exchanger network design process/hen.js

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• Choosing the next match

  Which hot and cold stream are matched next: streams at the pinch first, then the largest duty.

  score = 1e9 × (streams at the pinch) + min(remaining hot, remaining cold)

  Heat-exchanger network design process/hen.js

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• Duty and end temperatures of a match

  The heat a match exchanges, the temperatures of both streams at its two ends and whether it keeps the minimum temperature difference.

  Q = min(remaining hot, remaining cold); ΔT_hot = Q ÷ CP_hot; ΔT_cold = Q ÷ CP_cold; d₁ = |T_hot,₀ − T_cold,₀|; d₂ = |T_hot,₁ − T_cold,₁|

  Heat-exchanger network design process/hen.js

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• Heaters, coolers and totals

  The heating and cooling the matches leave over, and the number of units against the minimum.

  Q_H = Σ heater duties; Q_C = Σ cooler duties; units = matches + heaters + coolers

  Heat-exchanger network design process/hen.js

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• Numbers written on the grid drawing

  The CP, temperatures and duties shown as text on the network grid.

  label = value rounded to 1 decimal

  Heat-exchanger network design process/hen.js

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• Energy, cost and CO2 of a variant

  The totals a variant is judged by: heating, cooling and shaft power in kW, operating cost per hour and kg CO2 per hour.

  cost/h = (heat × p_heat + cool × p_cool) × 3600 ÷ 1e9 + (shaft ÷ 1e3) × p_power; CO2/h = (heat × 3600 ÷ 1e9) × f_heat + (shaft ÷ 1e3) × f_power

  Flowsheet variant metrics process/optimise.js

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• Proposed composition from lab results

  The mole fractions a sample's results give for a stream, for a person to accept or reject.

  w = % ÷ 100; w = ppm × 1e-6; w = (mg/L × 1e-3) ÷ ρ; nᵢ = wᵢ ÷ MWᵢ; zᵢ = nᵢ ÷ Σ n

  Lab results to stream composition process/env/lims.js

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• Calculated composition for the lab system

  A stream's composition as mass or mole fractions, percent or ppm, to send back as calculated values.

  wᵢ = zᵢ × MWᵢ ÷ Σ(z × MW); mass % = w × 100; ppm = w × 1e6

  Lab results to stream composition process/env/lims.js

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• Rational tag value

  A TIFF RATIONAL (two 32-bit integers) as one number.

  value = numerator ÷ denominator

  TIFF scan resolution raster/tiff.js

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• Resolution in dots per inch

  The horizontal and vertical resolution of a page, in dots per inch.

  dpi = resolution × 2.54 (centimeter files); dpi = resolution (inch files)

  TIFF scan resolution raster/tiff.js

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• Triangulate a polygon with holes

  The triangles of a polygon, as indexes into its point list.

  Polygon triangulation (ear clipping) usd/earcut.js

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• Winding of a ring

  Whether a ring runs clockwise or counterclockwise.

  sum = Σ (xⱼ − xᵢ) × (yᵢ + yⱼ)

  Polygon triangulation (ear clipping) usd/earcut.js

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• Orientation of three points

  On which side of the line through two points a third point lies.

  area = (q.y − p.y) × (r.x − q.x) − (q.x − p.x) × (r.y − q.y)

  Polygon triangulation (ear clipping) usd/earcut.js

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• Removing duplicate and collinear points

  Drops points that repeat their neighbor or lie on the line between their neighbors.

  Polygon triangulation (ear clipping) usd/earcut.js

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• Ear test

  Whether a corner can be cut off as a triangle: it is convex and no other point of the ring lies inside it.

  Polygon triangulation (ear clipping) usd/earcut.js

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• Point in triangle

  Whether a point lies inside or on a triangle.

  three cross products of (corner − point) pairs, each ≥ 0

  Polygon triangulation (ear clipping) usd/earcut.js

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• Bridge from a hole to the outer ring

  The point of the outer ring a hole is connected to.

  x = p.x + (h.y − p.y) × (p′.x − p.x) ÷ (p′.y − p.y); tan = |h.y − p.y| ÷ (h.x − p.x)

  Polygon triangulation (ear clipping) usd/earcut.js

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• Do two segments cross

  Whether two edges intersect, including touching and collinear overlap.

  Polygon triangulation (ear clipping) usd/earcut.js

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• Is the middle of a diagonal inside the polygon

  Whether a diagonal between two corners runs through the inside of the polygon.

  midpoint = ((a.x + b.x) ÷ 2, (a.y + b.y) ÷ 2); even-odd count of edges crossed by a ray

  Polygon triangulation (ear clipping) usd/earcut.js

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• Value of a number row

  The number handed to the app when a person changes a number box.

  v = min(max, max(min, typed value))

  Numbers typed into the property grid app-propgrid.js Every app
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• Rounding of a slider value

  Removes binary noise such as 0.30000000000000004 from a slider's value.

  v = round to (decimals of the step + 2) decimals, at most 8

  Numbers typed into the property grid app-propgrid.js Every app
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• Value of a slider row

  The number handed to the app while a slider is dragged or its box is changed.

  v = fix(clamp(value, min, max), step)

  Numbers typed into the property grid app-propgrid.js Every app
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• Integer or float

  Decides how a number is written.

  Numbers sent between people editing together live-client.js Every app
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• Writing a float

  Writes a number as 8 bytes.

  Numbers sent between people editing together live-client.js Every app
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• Writing an integer

  Writes a whole number in the shortest integer form that holds it.

  Numbers sent between people editing together live-client.js Every app
  Exact

• Reading a float

  Reads a 4-byte or an 8-byte float from a message.

  Numbers sent between people editing together live-client.js Every app
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• Reading a 64-bit integer

  Reads an unsigned or signed 64-bit integer.

  Numbers sent between people editing together live-client.js Every app
  Exact

• Cost of one call

  The cost of a call from its token counts and the model's prices.

  cost = (in × price in + out × price out) ÷ 1,000,000

  Spending on a person's own AI model ai-byok.js Every app
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• Spending today and this month

  The cost, tokens and number of calls for the current day and month, and the calls in the last hour.

  amount = Σ cost of the calls in the period

  Spending on a person's own AI model ai-byok.js Every app
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• Check before a call

  Refuses a call when the calls per hour, the day's or month's spending, or the day's tokens could pass a limit.

  refuse when spent + worst-case cost > limit; worst case = cost(estimated input, largest allowed output)

  Spending on a person's own AI model ai-byok.js Every app
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• Recorded cost and the token estimate

  Stores each call's cost and estimates the input tokens of the next request.

  estimated tokens = ⌈JSON length ÷ 3⌉ + 500

  Spending on a person's own AI model ai-byok.js Every app
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• Flow through a passage

  Persons per second through a door or corridor of a given width.

  rate = max(0, Fsm × (width − 2 × boundary))

  People flow and evacuation time people.js Every app
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• Speed and specific flow at a density

  Walking speed and flow per meter of width at a given density of people.

  S = k × (1 − 0.266 × D); Fs = S × D; k = 1.4 m/s

  People flow and evacuation time people.js Every app
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• Rate and speed of each passage

  The flow rate and walking speed each passage gets, from its own values or the defaults for its kind.

  People flow and evacuation time people.js Every app
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• Door service time and walking time

  How long a door takes per person, how long the passage takes to walk, and each room's distance to an exit in time.

  service = 1 ÷ rate; walk = length ÷ speed; to exit = least total walking time

  People flow and evacuation time people.js Every app
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• Which way a room empties

  The next passage for people in a room: the one with the least remaining walking time to an exit.

  keep a hop when toExit(next) < toExit(room) − 1e-9; choose the least walk + toExit(next)

  People flow and evacuation time people.js Every app
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• Evacuation results

  People evacuated, the time the last person reaches an exit, peak occupancy and density per room, and queues per door.

  time = latest arrival at any exit; density = peak people in a room ÷ its area

  People flow and evacuation time people.js Every app
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• Joint frames at a pose

  Each joint's rotation and offset from its parent at the given joint values.

  Rigid-body dynamics of a mechanism dynamics.js Every app
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• Joint torques for a motion (inverse dynamics)

  The torque or force each joint needs for given positions, velocities and accelerations, with gravity, external loads and damping.

  outward: ω, α, a of each link; F = m × a_com; N = I α + ω × (I ω); inward: τᵢ = axis · n (revolute) or axis · f (prismatic) + damping × q̇ᵢ

  Rigid-body dynamics of a mechanism dynamics.js Every app
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• Mass matrix

  The joint-space mass matrix at a pose.

  column j = rnea(q, q̇ = 0, q̈ = unit vector j) with no gravity, damping or external loads

  Rigid-body dynamics of a mechanism dynamics.js Every app
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• Joint accelerations for given torques (forward dynamics)

  The acceleration of each joint under applied torques.

  M q̈ = τ − h

  Rigid-body dynamics of a mechanism dynamics.js Every app
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• Kinetic and potential energy

  The mechanism's kinetic, potential and total energy at a pose and velocity.

  KE = ½ q̇ᵀ M q̇; PE = −Σ m × (g · p_com)

  Rigid-body dynamics of a mechanism dynamics.js Every app
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• Simulated motion over time

  Joint positions and velocities at each time step under gravity and applied torques.

  RK4: y₊ = y + dt ÷ 6 × (k₁ + 2k₂ + 2k₃ + k₄); steps = round(T ÷ dt); t = step × dt

  Rigid-body dynamics of a mechanism dynamics.js Every app
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• Courtyard of a footprint

  The rectangle around all pads, grown by a margin.

  courtyard = (min(x − w ÷ 2) − m, min(y − h ÷ 2) − m, max(x + w ÷ 2) + m, max(y + h ÷ 2) + m)

  Parametric PCB footprints footlib.js Every app
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• Two-terminal chip

  Two pads at half the pitch either side of the center.

  x = ∓ pitch ÷ 2

  Parametric PCB footprints footlib.js Every app
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• Dual-row surface-mount package

  Pads in two rows, numbered counter-clockwise from the top left.

  y = ((n ÷ 2 − 1) ÷ 2 − i) × pitch; x = ∓ row spacing ÷ 2

  Parametric PCB footprints footlib.js Every app
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• Dual in-line through-hole package

  Drilled pads in two rows.

  y = ((n ÷ 2 − 1) ÷ 2 − i) × pitch; x = ∓ row spacing ÷ 2

  Parametric PCB footprints footlib.js Every app
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