Calculations

Calculations

• Double with default, read

  A value stored as the bytes in which it differs from a default value.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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  64-bit

• 64-bit whole number, read

  An unsigned 64-bit whole number from two 32-bit halves.

  value = high × 2³² + low

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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  64-bit

• Bit long long, read

  A whole number of up to seven bytes, the count given by three bits.

  value = Σ byteᵢ × 256ⁱ

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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  64-bit

• Handle reference, read

  An object handle, absolute or as an offset from another handle.

  value = Σ byte × 256ᵏ; codes 6, 8, A, C: ref + 1, ref − 1, ref + value, ref − value

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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  64-bit

• Modular char, read

  A signed whole number stored seven bits per byte.

  value = Σ (byte & 0x7F) × 2^(7i); the last byte carries 6 bits and the sign

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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  64-bit

• Raw double, written

  A 64-bit value written as its eight bytes.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  64-bit

• Bit double, written

  A value written as a two-bit code for exactly 0.0 or 1.0, or as a full 64-bit value.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  64-bit

• Double with default, written

  A value written as a two-bit code when it equals its default, or in full.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  64-bit

• Thickness, written

  A thickness written as one bit when it is zero, or as a bit double.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  64-bit

• Bit long, written

  A 32-bit whole number in its short or full form.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  Exact

• Bit short, written

  A 16-bit whole number in its short or full form.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  Exact

• Bit long long, written

  A 64-bit whole number as a three-bit byte count and its bytes.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  Exact

• Point, written

  The three coordinates of a point as bit doubles.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  64-bit

• Handle bytes, written

  A handle as its bytes, most significant first.

  byte = v mod 256; v = ⌊v ÷ 256⌋

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  Exact

• Lineweight to stored index

  The index written for a lineweight, or the codes for by layer, by block and default.

  index of the list value with the smallest |value − lineweight|

  Lineweights in a DWG file dwg/model.js

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  Exact

• Stored index to lineweight

  The lineweight for an index read from a file.

  Lineweights in a DWG file dwg/model.js

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  Exact

• Variant changed to another type

  The value of a variant as a string, a double, a single, an integer, a long or a boolean.

  integer = round(n); double = n

  Number conversion in LISP variants lisp/vla.js

  • CadCraft
  64-bit

• Longest clear part of a move

  The part of a straight move that stays clear of other solids.

  steps = min(160, max(4, ⌈|δ| ÷ (thin ÷ 2)⌉)); then 24 halvings between the last clear and the first blocked sample

  Where a moved solid stops against another m3d/collide.js

  • CadCraft
  • ModelCraft
  64-bit

• Thinnest dimension

  The smallest box side of the moving pieces and of everything in the way, which sets the sample spacing.

  thin = max(0.005, min over boxes of the smallest side above 1e-4)

  Where a moved solid stops against another m3d/collide.js

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  64-bit

• Edge through the inside of a face

  Whether a segment passes through a face's interior.

  sₚ = n·p − d, s_q = n·q − d; t = sₚ ÷ (sₚ − s_q)

  Where a moved solid stops against another m3d/collide.js

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  64-bit

• Point in a face

  Whether a point on a face's plane lies in one of its triangles.

  u, v from cross products ÷ |n|²; inside when u, v, 1 − u − v ≥ −1e-9

  Where a moved solid stops against another m3d/collide.js

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  64-bit

• Point inside a solid

  Whether a point lies inside the closed surface made of an object's faces.

  Where a moved solid stops against another m3d/collide.js

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  64-bit

• Ray against a triangle

  The distance along a ray to a triangle, or nothing when it misses.

  det = e₁·(d × e₂); u = (s·(d × e₂)) ÷ det; v = (d·(s × e₁)) ÷ det; t = (e₂·(s × e₁)) ÷ det

  Where a moved solid stops against another m3d/collide.js

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  64-bit

• Box of a placed component

  The bounding box in the model of a component's faces and edges, from its own coordinates and its placement matrix.

  X = (m₀x + m₄y + m₈z + m₁₂) ÷ w

  Where a moved solid stops against another m3d/collide.js

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  64-bit

• Triangles of a polygon with holes

  The triangles of a face as triples of vertex indexes.

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Orientation of three points

  Twice the signed area of a triangle, whose sign says which way three points turn.

  (q_y − p_y)(r_x − q_x) − (q_x − p_x)(r_y − q_y)

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Direction of a ring

  Whether a ring of points runs clockwise or counterclockwise.

  Σ (x_j − x_i)(y_i + y_j)

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Is a corner an ear

  Whether the triangle at a corner is convex and holds no other point of the polygon.

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Point in a triangle

  Whether a point lies in or on a triangle.

  three comparisons (c − p) × (a − p) ≥ 0, each written as product ≥ product

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Duplicate and collinear points removed

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

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Bridge from a hole to the outer ring

  The outer-ring point a hole is connected to, found by a ray to the left from the hole's leftmost point.

  x = p_x + (h_y − p_y)(n_x − p_x) ÷ (n_y − p_y); tan = |h_y − p_y| ÷ (h_x − p_x)

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Do two edges cross

  Whether two segments intersect, used to keep a diagonal from crossing the polygon.

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Is a diagonal's midpoint inside

  Whether the middle of a candidate diagonal lies inside the polygon.

  crossing count of a ray from ((a_x + b_x) ÷ 2, (a_y + b_y) ÷ 2)

  Triangulation of faces with holes m3d/earcut.js

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  64-bit

• Start and finish of each step

  When each step of each batch starts and finishes, and on which vessel.

  release = start + b × interval; t₀ = max(ready, vessel free); t₁ = t₀ + duration; wait = t₀ − ready

  Batch schedule and batch inventory process/batch.js

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  64-bit

• Makespan, utilization, cycle time and batches per day

  The length of the whole schedule, how busy each vessel is, the steady cycle time and the batches per day it allows.

  makespan = max finish − start; utilization = Σ(end − start) ÷ makespan; cycle = (work + clean × uses) ÷ vessels; batches per day = 24 ÷ cycle

  Batch schedule and batch inventory process/batch.js

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  64-bit

• Hours to whole days for the Gantt view

  The start and finish day of each task for the schedule view, which works in whole days.

  start day = day0 + ⌊start ÷ 24 + 1e-9⌋; finish day = day0 + max(⌊(finish − 1e-9) ÷ 24⌋, ⌊start ÷ 24 + 1e-9⌋)

  Batch schedule and batch inventory process/batch.js

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  64-bit

• Batch mole inventory and atom balance

  The moles of each component in each vessel after every operation, what was discharged, and how well the atoms balance.

  transfer: q = n × fraction; react: ξ = conversion × n_key ÷ |ν_key|, n ← n + ν × ξ; atom error = max |charged − held| ÷ charged

  Batch schedule and batch inventory process/batch.js

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  64-bit

• Monod growth in a batch or fed-batch vessel

  Concentrations of biomass, substrate and product, the volume, and their masses at each output time.

  μ = μmax × S ÷ (Ks + S); dX/dt = (μ − kd) X − D X; dS/dt = −(μ ÷ Yxs + m) X + D (Sf − S); dP/dt = (α μ + β) X − D P; dV/dt = F; D = F ÷ V

  Fermentation (Monod kinetics) process/batch.js

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  64-bit

• Channel counts and counts with spare

  How many analog and digital inputs and outputs the I/O list needs, and how many once the spare allowance is added.

  with spare = ⌈n × (1 + spare)⌉

  I/O channel counts with spare allowance process/controls.js

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  64-bit

• Lower and higher heating value

  The heat released by burning one mole of a component, with the water formed as vapor or as liquid.

  LHV = h_f − n_C × h_f(CO2) − (n_H ÷ 2) × h_f(H2O gas); HHV uses h_f(H2O liquid)

  Process emissions (CO2 equivalent) process/emissions.js

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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit