wikiluntti - User contributions [en]

Topography of a region - Python

2026-04-06T13:46:00Z

Mol: /* Introduction */

== Introduction ==

https://www.norgeskart.no/?backgroundLayer=toporaster&lon=822368.0472314445&lat=7767892.806770398&rotation=0&zoom=7&printTool=hiking

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

*DEM Digital elevation model
*DSM Digital surface model
*DTM Digital terrain model

=== Python GIS ===

* folium
* geopy
* GeoPandas

== Earth Model ==

DEM files (likes SRTM data) with
* Rasterio
* GDAL

Norwegian national DEM (Kartverket). The Norwegian Mapping Authority provides very high-resolution DEM data (down to 1 meter). https://hoydedata.no. The files in the .zip package
* .tif: pixel = elevation value
* .tfw (world file). Georeferencing info (pixel size, rotation, coordinates)
* .tif.aux.xml. May include Statistics (min/max elevation), Pyramid layers (for faster display), Projection overrides
* .prj Defines the coordinate reference system (CRS)
* .cpg. character encoding
* .shp. Contains shapes (points, lines, or polygons)
* .shx. Shape index file
* .dbf. Attribute table (like a spreadsheet)

Global DEMs (digital elevation model). No need super high resolution. Common datasets:
* SRTM (~30 m resolution)
* ASTER GDEM (~30 m)

Open portals. You can use
* OpenTopography
* OpenDEM
* NASA EarthData
which gives eg GeoTIFF.

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== Python and GeoTIFF ==

<syntaxhighlight lang="python">
import rasterio

with rasterio.open("dem.tif") as dem:
for lon, lat in track:
row, col = dem.index(lon, lat)
elevation = dem.read(1)[row, col]
</syntaxhighlight>

Downsampling the GeoTIFF
<syntaxhighlight lang="python">
from rasterio.enums import Resampling

data = src.read(
out_shape=(
src.count,
src.height // 10,
src.width // 10
),
resampling=Resampling.average
)
</syntaxhighlight>

== Path-finding ==

A depression filling algorithm, such as:
* Planchon–Darboux algorithm
* Wang & Liu algorithm
* Priority-flood (modern, very efficient). Overflow-hydro or RichDEM, WhoteboxTools.

Flow direction using:
* D8 flow direction (simplest, most common)
* D∞ (more accurate, distributes flow)
* MFD (multiple flow direction)

Treat the DEM as a graph. Nodes are raster cells and edges neighbor connections. The Cost function can be
# Elevation drop (prefer downhill)
# Slope
# Accumulated flow (prefer existing rivers)
And use Dijkstra’s algorithm or A* (if you know destination).

=== Tools ===

GDAL (basic raster ops)
TauDEM (excellent for hydrology)
WhiteboxTools (very strong, modern)
GRASS GIS (r.watershed, r.fill.dir)

=== Priority-flood Python ===

#Start from the edges (they always drain outward)
#Use a priority queue (min-heap) ordered by elevation
#Expand inward
#If a neighbor is lower → raise it (fill the sink)
#Otherwise → keep it as-is

<syntaxhighlight lang="python">
import heapq
import numpy as np

def priority_flood(dem):
"""
Fill depressions in a DEM using Priority-Flood.

Parameters:
dem (2D numpy array): elevation grid

Returns:
filled_dem (2D numpy array)
"""
rows, cols = dem.shape
filled = dem.copy()
visited = np.zeros((rows, cols), dtype=bool)

heap = []

# Step 1: push all edge cells into heap
for r in range(rows):
for c in [0, cols - 1]:
heapq.heappush(heap, (filled[r, c], r, c))
visited[r, c] = True

for c in range(cols):
for r in [0, rows - 1]:
if not visited[r, c]:
heapq.heappush(heap, (filled[r, c], r, c))
visited[r, c] = True

# 4-neighborhood (can switch to 8 if needed)
neighbors = [(-1, 0), (1, 0), (0, -1), (0, 1)]

# Step 2: process cells
while heap:
elev, r, c = heapq.heappop(heap)

for dr, dc in neighbors:
nr, nc = r + dr, c + dc

if 0 <= nr < rows and 0 <= nc < cols and not visited[nr, nc]:
visited[nr, nc] = True

# If neighbor is lower → fill it
if filled[nr, nc] < elev:
filled[nr, nc] = elev

heapq.heappush(heap, (filled[nr, nc], nr, nc))

return filled
</syntaxhighlight>

== 11 ==

Topography of a region - Python

2026-04-05T17:14:50Z

Mol: /* Earth Model */

== Introduction ==

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

*DEM Digital elevation model
*DSM Digital surface model
*DTM Digital terrain model

=== Python GIS ===

* folium
* geopy
* GeoPandas

== Earth Model ==

DEM files (likes SRTM data) with
* Rasterio
* GDAL

Norwegian national DEM (Kartverket). The Norwegian Mapping Authority provides very high-resolution DEM data (down to 1 meter). https://hoydedata.no. The files in the .zip package
* .tif: pixel = elevation value
* .tfw (world file). Georeferencing info (pixel size, rotation, coordinates)
* .tif.aux.xml. May include Statistics (min/max elevation), Pyramid layers (for faster display), Projection overrides
* .prj Defines the coordinate reference system (CRS)
* .cpg. character encoding
* .shp. Contains shapes (points, lines, or polygons)
* .shx. Shape index file
* .dbf. Attribute table (like a spreadsheet)

Global DEMs (digital elevation model). No need super high resolution. Common datasets:
* SRTM (~30 m resolution)
* ASTER GDEM (~30 m)

Open portals. You can use
* OpenTopography
* OpenDEM
* NASA EarthData
which gives eg GeoTIFF.

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== Python and GeoTIFF ==

<syntaxhighlight lang="python">
import rasterio

with rasterio.open("dem.tif") as dem:
for lon, lat in track:
row, col = dem.index(lon, lat)
elevation = dem.read(1)[row, col]
</syntaxhighlight>

Downsampling the GeoTIFF
<syntaxhighlight lang="python">
from rasterio.enums import Resampling

data = src.read(
out_shape=(
src.count,
src.height // 10,
src.width // 10
),
resampling=Resampling.average
)
</syntaxhighlight>

== Path-finding ==

A depression filling algorithm, such as:
* Planchon–Darboux algorithm
* Wang & Liu algorithm
* Priority-flood (modern, very efficient). Overflow-hydro or RichDEM, WhoteboxTools.

Flow direction using:
* D8 flow direction (simplest, most common)
* D∞ (more accurate, distributes flow)
* MFD (multiple flow direction)

Treat the DEM as a graph. Nodes are raster cells and edges neighbor connections. The Cost function can be
# Elevation drop (prefer downhill)
# Slope
# Accumulated flow (prefer existing rivers)
And use Dijkstra’s algorithm or A* (if you know destination).

=== Tools ===

GDAL (basic raster ops)
TauDEM (excellent for hydrology)
WhiteboxTools (very strong, modern)
GRASS GIS (r.watershed, r.fill.dir)

=== Priority-flood Python ===

#Start from the edges (they always drain outward)
#Use a priority queue (min-heap) ordered by elevation
#Expand inward
#If a neighbor is lower → raise it (fill the sink)
#Otherwise → keep it as-is

<syntaxhighlight lang="python">
import heapq
import numpy as np

def priority_flood(dem):
"""
Fill depressions in a DEM using Priority-Flood.

Parameters:
dem (2D numpy array): elevation grid

Returns:
filled_dem (2D numpy array)
"""
rows, cols = dem.shape
filled = dem.copy()
visited = np.zeros((rows, cols), dtype=bool)

heap = []

# Step 1: push all edge cells into heap
for r in range(rows):
for c in [0, cols - 1]:
heapq.heappush(heap, (filled[r, c], r, c))
visited[r, c] = True

for c in range(cols):
for r in [0, rows - 1]:
if not visited[r, c]:
heapq.heappush(heap, (filled[r, c], r, c))
visited[r, c] = True

# 4-neighborhood (can switch to 8 if needed)
neighbors = [(-1, 0), (1, 0), (0, -1), (0, 1)]

# Step 2: process cells
while heap:
elev, r, c = heapq.heappop(heap)

for dr, dc in neighbors:
nr, nc = r + dr, c + dc

if 0 <= nr < rows and 0 <= nc < cols and not visited[nr, nc]:
visited[nr, nc] = True

# If neighbor is lower → fill it
if filled[nr, nc] < elev:
filled[nr, nc] = elev

heapq.heappush(heap, (filled[nr, nc], nr, nc))

return filled
</syntaxhighlight>

== 11 ==

Topography of a region - Python

2026-04-05T13:04:42Z

Mol: /* Priority-flood Python */

== Introduction ==

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

*DEM Digital elevation model
*DSM Digital surface model
*DTM Digital terrain model

=== Python GIS ===

* folium
* geopy
* GeoPandas

== Earth Model ==

DEM files (likes SRTM data) with
* Rasterio
* GDAL

Norwegian national DEM (Kartverket). The Norwegian Mapping Authority provides very high-resolution DEM data (down to 1 meter).
* https://hoydedata.no

Global DEMs (digital elevation model). No need super high resolution. Common datasets:
* SRTM (~30 m resolution)
* ASTER GDEM (~30 m)

Open portals. You can use
* OpenTopography
* OpenDEM
* NASA EarthData
which gives eg GeoTIFF.

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== Python and GeoTIFF ==

<syntaxhighlight lang="python">
import rasterio

with rasterio.open("dem.tif") as dem:
for lon, lat in track:
row, col = dem.index(lon, lat)
elevation = dem.read(1)[row, col]
</syntaxhighlight>

Downsampling the GeoTIFF
<syntaxhighlight lang="python">
from rasterio.enums import Resampling

data = src.read(
out_shape=(
src.count,
src.height // 10,
src.width // 10
),
resampling=Resampling.average
)
</syntaxhighlight>

== Path-finding ==

A depression filling algorithm, such as:
* Planchon–Darboux algorithm
* Wang & Liu algorithm
* Priority-flood (modern, very efficient). Overflow-hydro or RichDEM, WhoteboxTools.

Flow direction using:
* D8 flow direction (simplest, most common)
* D∞ (more accurate, distributes flow)
* MFD (multiple flow direction)

Treat the DEM as a graph. Nodes are raster cells and edges neighbor connections. The Cost function can be
# Elevation drop (prefer downhill)
# Slope
# Accumulated flow (prefer existing rivers)
And use Dijkstra’s algorithm or A* (if you know destination).

=== Tools ===

GDAL (basic raster ops)
TauDEM (excellent for hydrology)
WhiteboxTools (very strong, modern)
GRASS GIS (r.watershed, r.fill.dir)

=== Priority-flood Python ===

#Start from the edges (they always drain outward)
#Use a priority queue (min-heap) ordered by elevation
#Expand inward
#If a neighbor is lower → raise it (fill the sink)
#Otherwise → keep it as-is

<syntaxhighlight lang="python">
import heapq
import numpy as np

def priority_flood(dem):
"""
Fill depressions in a DEM using Priority-Flood.

Parameters:
dem (2D numpy array): elevation grid

Returns:
filled_dem (2D numpy array)
"""
rows, cols = dem.shape
filled = dem.copy()
visited = np.zeros((rows, cols), dtype=bool)

heap = []

# Step 1: push all edge cells into heap
for r in range(rows):
for c in [0, cols - 1]:
heapq.heappush(heap, (filled[r, c], r, c))
visited[r, c] = True

for c in range(cols):
for r in [0, rows - 1]:
if not visited[r, c]:
heapq.heappush(heap, (filled[r, c], r, c))
visited[r, c] = True

# 4-neighborhood (can switch to 8 if needed)
neighbors = [(-1, 0), (1, 0), (0, -1), (0, 1)]

# Step 2: process cells
while heap:
elev, r, c = heapq.heappop(heap)

for dr, dc in neighbors:
nr, nc = r + dr, c + dc

if 0 <= nr < rows and 0 <= nc < cols and not visited[nr, nc]:
visited[nr, nc] = True

# If neighbor is lower → fill it
if filled[nr, nc] < elev:
filled[nr, nc] = elev

heapq.heappush(heap, (filled[nr, nc], nr, nc))

return filled
</syntaxhighlight>

== 11 ==

Topography of a region - Python

2026-04-04T10:44:59Z

Mol: /* 11 */

Topography of a region - Python

2026-04-04T10:42:24Z

Mol: /* 11 */

Topography of a region - Python

2026-04-04T10:23:45Z

Mol: /* Introduction */

Topography of a region - Python

2026-04-04T10:18:19Z

Mol: /* Earth Model */

== Introduction ==

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

=== Python GIS ===

* folium
* geopy
* GeoPandas

== Earth Model ==

DEM files (likes SRTM data) with
* Rasterio
* GDAL

Norwegian national DEM (Kartverket). The Norwegian Mapping Authority provides very high-resolution DEM data (down to 1 meter).
* https://hoydedata.no

Global DEMs (digital elevation model). No need super high resolution. Common datasets:
* SRTM (~30 m resolution)
* ASTER GDEM (~30 m)

Open portals. You can use
* OpenTopography
* OpenDEM
* NASA EarthData
which gives eg GeoTIFF.

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== 11 ==

== 11 ==

== 11 ==

Topography of a region - Python

2026-04-04T10:16:02Z

Mol: /* Earth Model */

== Introduction ==

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

=== Python GIS ===

* folium
* geopy
* GeoPandas

== Earth Model ==

DEM files (likes SRTM data) with
* Rasterio
* GDAL

Norwegian national DEM (Kartverket). The Norwegian Mapping Authority provides very high-resolution DEM data (down to 1 meter).
* https://hoydedata.no

Global DEMs (quick + easy). No need super high resolution. Common datasets:
* SRTM (~30 m resolution)
* ASTER GDEM (~30 m)

Open portals. You can use
* OpenTopography
* OpenDEM
* NASA EarthData
which gives eg GeoTIFF.

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== 11 ==

== 11 ==

== 11 ==

Topography of a region - Python

2026-04-04T10:12:50Z

Mol: /* Earth Model */

== Introduction ==

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

=== Python GIS ===

* folium
* geopy
* GeoPandas

== Earth Model ==

DEM files (likes SRTM data) with
* Rasterio
* GDAL

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== 11 ==

== 11 ==

== 11 ==

Topography of a region - Python

2026-04-04T10:10:40Z

Mol: /* Python GIS */

Topography of a region - Python

2026-04-04T10:06:18Z

Mol: /* Open-Elevation.con */

== Introduction ==

An example of Alta-joki Alta river.

https://caltopo.com/map.html#ll=68.8546,23.64807&z=8&b=mbt

Mostly needs an API.

=== Python GIS ===

geopy

== Earth Model ==

== Open Topo Data ==

https://www.opentopodata.org/

== Open-Elevation.com ==

== 11 ==

== 11 ==

== 11 ==

Topography of a region - Python

2026-04-04T10:06:07Z

Mol: /* 11 */

Mol: /* Real gas: Air */

== Introduction ==

=== 1 ===

<gallery>
DieselCycle pvDiagram simple.png|Diesel Cycle
DieselCycle.gif|Diesel Cycle
</gallery>

Ratio of specific heats (heat capacity ratio) is defined as
<math>
\gamma = \frac{ C_p }{ C_v }
</math>

=== pV diagram ===

# Isentropic (adiabatic) expansion
# Isochoric cooling (Qout): Heat rejection. Power stroke ends, heat rejection starts.
# Isobaric compression: Exhaust
# Isobaric expansion: Intake
# Isentropic (adiabatic) compression
# Isobaric heating (Qin): Combustion of fuel (heat is added in a constant pressure;)

'''Engine displacement''' is the cylinder volume swept by all of the pistons of a piston engine, excluding the combustion chambers. A '''combustion chamber''' is part of an internal combustion engine in which the fuel/air mix is burned.

Only air is compressed, and then diesel fuel is injected directly into that hot, high-pressure air.
* Cylinder pressure: ~30–80 bar
* Injection pressure: ~1,000–2,500+ bar

=== Diesel Cycle and Ideal Gas ===

For a closed system, the total change in energy of a system is the sum of the work done and the heat added <math>dU = \delta W + \delta Q</math>, and the reversible work done on a system by changing the volume is <math>\delta W = - p dV</math>.

If the system is reversible and adiabatic (isentropic) <math>\delta Q = 0</math>, which gives

<math>
dU = \delta W + \delta Q = -pdV + 0
</math>

Furthermore, for any transformation of an ideal gas, it is always true that <math>dU = nC_v dT</math>, giving

<math>
dU = nC_v dT = -pdV
</math>

=== Real gas: Air ===

Basic ideal gas model (good first approximation). Air is usually approximated as an ideal gas with:
* Gas constant:
** R≈287 J/(kg\cdotpK)
** R≈287J/(kg\cdotpK)
* Equation of state:
* p=ρRT
This works well at:
* pressures near atmospheric
* temperatures roughly 200–500 K

“Generalized” ideal gas → temperature-dependent properties. To go beyond the simple model, you allow properties like heat capacity to vary with temperature:

Heat capacity. For air, a common approximation is a polynomial
<math>
c_p(T)=a+bT.
</math>
Typical coefficients (for dry air, ~200–1000 K range):
* a≈1005
* b≈0.1
More accurate forms come from NASA polynomials:
<math>
\frac{c_p}R=a_1+a_2T+a_3T^2+a_4T^3+a_5T^4
</math>
Which are widely used in CFD and thermodynamics.

Compressibility factor Z (real gas correction). If you want a generalized ideal gas, you often introduce <math>p=Z\rho RT</math>. Where
* Z=1 → ideal gas
* Z≠1 → real gas behavior
For air:
* At normal conditions: Z≈1
* At high pressure: use virial expansion
<math>
Z=1+\frac{B(T)}V+\frac{C(T)}{V^2}+ \cdots
\approx 1+ \frac{B(T)}{RT}p
</math>

Mixture-based formulation (more fundamental). Air is a mixture mainly of:
* N₂ (~78%)
* O₂ (~21%)
* Ar (~1%)

<math>
\begin{align}
R &=\sum_i y_i R_i \\
c_p & = \sum_i y_ i c_{p,i}(T)
\end{align}
</math>

== Realistic Diesel Cycle ==

<gallery>
Cylinder.png| The size of the combustion chamber of MB W211.
</gallery>

=== Ratio of specific heats γ ===

<math>
\gamma = \frac{ \sum y_1 c_{p,i} }{ \sum y_1 ( c_{p,i} -R_u ) }
</math>

{| class="wikitable"
|+ Caption text
|-
! Condition !! γ (approx.)
|-
| Stoichiometric || ~1.30–1.33
|-
| Moderate lean || ~1.33–1.37
|-
| Very lean (diesel) || ~1.37–1.40
|}

=== Injection pressures ===

* Older systems: 200–500 bar
* Modern common-rail: 1,000–2,500 bar
* Latest systems: up to ~3,000 bar

* Air pressure in cylinder: ~30–80 bar, which is the pressure of the combustion chamber.
* Temperature: ~700–1000 K

=== Diesel-air Mixture ===

Only diesel is ejected into the cylinder. Clean air comes in during the intake stroke.

The heat capacity ratio (known as the adiabatic index) for a diesel-air mixture is typically around 1.4.

{| class="wikitable"
|+ Air-fuel ratio
|-
| 14.5:1 || Near-stoichiometric; good combustion efficiency but higher emissions.
|-
| 16:1 || Balanced performance; good power output and efficiency.
|-
| 18:1 || Lean burn; improved fuel economy but potential for higher NOx emissions.
|}

Diesel is a complicated compound, but it’s commonly approximated as a hydrocarbon like C12H23 (or C12H26). Air is about 21% O2 and 79% N2. Without nitrogen, the stoichiometric ratio is about

C12H23+17.75O2→12CO2+11.5H2O

and by including nitrogen, we get

C12H23+17.75( O2 + 3.76 N2 )→12CO2+11.5H2O + 66.74 N2.

The molar masses
* Fuel: 167 g/mol
* Air: 137.28 g/mol
And the total air needed is 17.72 x 137.28 = 2436 g, which gives air-to-fuel-ratio to

<math>
AFR = \frac{2436}{167} = 14.6:1
</math>

=== MB data ===

Motor: OM648.

Mercedes Benz W211 (2003)
* Engine displacement: 3222 cm3 = 0.003222 m3
* Bore x Stroke: 88.0 x 88.4 mm3
* Compression Ratio: 18.0

Bore x stroke gives V = 6xπ(8.8/2)2 x 8.84 cm3 = 3225.95cm3, which is rather close.

<math>
p_1 V_1^\gamma = \text{constant}_1
</math>

=== 1 ===

=== 1 ===

=== 1 ===

Diesel Cycle

2026-03-30T17:58:29Z

Mol: /* Real gas: Air */

== Introduction ==

=== 1 ===

<gallery>
DieselCycle pvDiagram simple.png|Diesel Cycle
DieselCycle.gif|Diesel Cycle
</gallery>

Ratio of specific heats (heat capacity ratio) is defined as
<math>
\gamma = \frac{ C_p }{ C_v }
</math>

=== pV diagram ===

# Isentropic (adiabatic) expansion
# Isochoric cooling (Qout): Heat rejection. Power stroke ends, heat rejection starts.
# Isobaric compression: Exhaust
# Isobaric expansion: Intake
# Isentropic (adiabatic) compression
# Isobaric heating (Qin): Combustion of fuel (heat is added in a constant pressure;)

'''Engine displacement''' is the cylinder volume swept by all of the pistons of a piston engine, excluding the combustion chambers. A '''combustion chamber''' is part of an internal combustion engine in which the fuel/air mix is burned.

Only air is compressed, and then diesel fuel is injected directly into that hot, high-pressure air.
* Cylinder pressure: ~30–80 bar
* Injection pressure: ~1,000–2,500+ bar

=== Diesel Cycle and Ideal Gas ===

For a closed system, the total change in energy of a system is the sum of the work done and the heat added <math>dU = \delta W + \delta Q</math>, and the reversible work done on a system by changing the volume is <math>\delta W = - p dV</math>.

If the system is reversible and adiabatic (isentropic) <math>\delta Q = 0</math>, which gives

<math>
dU = \delta W + \delta Q = -pdV + 0
</math>

Furthermore, for any transformation of an ideal gas, it is always true that <math>dU = nC_v dT</math>, giving

<math>
dU = nC_v dT = -pdV
</math>

=== Real gas: Air ===

Basic ideal gas model (good first approximation). Air is usually approximated as an ideal gas with:
* Gas constant:
** R≈287 J/(kg\cdotpK)
** R≈287J/(kg\cdotpK)
* Equation of state:
* p=ρRT
This works well at:
* pressures near atmospheric
* temperatures roughly 200–500 K

“Generalized” ideal gas → temperature-dependent properties. To go beyond the simple model, you allow properties like heat capacity to vary with temperature:

Heat capacity. For air, a common approximation is a polynomial
<math>
c_p(T)=a+bT.
</math>
Typical coefficients (for dry air, ~200–1000 K range):
* a≈1005
* b≈0.1
More accurate forms come from NASA polynomials:
<math>
\frac{c_p}R=a_1+a_2T+a_3T^2+a_4T^3+a_5T^4
</math>
Which are widely used in CFD and thermodynamics.

Compressibility factor Z (real gas correction). If you want a generalized ideal gas, you often introduce <math>p=Z\rho RT</math>. Where
* Z=1 → ideal gas
* Z≠1 → real gas behavior
For air:
* At normal conditions: Z≈1
* At high pressure: use virial expansion
<math>
Z=1+\frac{B(T)}V+\frac{C(T)}{V^2}+ \cdots
\approx 1+ \frac{B(T)}{RT}p
</math>

Mixture-based formulation (more fundamental). Air is a mixture mainly of:
* N₂ (~78%)
* O₂ (~21%)
* Ar (~1%)

<math>
\begin{align*}
R &=\sum_i y_i R_i \\
c_p & = \sum_i y_ i c_{p,i}(T)
\end{align*}
</math>

== Realistic Diesel Cycle ==

<gallery>
Cylinder.png| The size of the combustion chamber of MB W211.
</gallery>

=== Ratio of specific heats γ ===

<math>
\gamma = \frac{ \sum y_1 c_{p,i} }{ \sum y_1 ( c_{p,i} -R_u ) }
</math>

{| class="wikitable"
|+ Caption text
|-
! Condition !! γ (approx.)
|-
| Stoichiometric || ~1.30–1.33
|-
| Moderate lean || ~1.33–1.37
|-
| Very lean (diesel) || ~1.37–1.40
|}

=== Injection pressures ===

* Older systems: 200–500 bar
* Modern common-rail: 1,000–2,500 bar
* Latest systems: up to ~3,000 bar

* Air pressure in cylinder: ~30–80 bar, which is the pressure of the combustion chamber.
* Temperature: ~700–1000 K

=== Diesel-air Mixture ===

Only diesel is ejected into the cylinder. Clean air comes in during the intake stroke.

The heat capacity ratio (known as the adiabatic index) for a diesel-air mixture is typically around 1.4.

{| class="wikitable"
|+ Air-fuel ratio
|-
| 14.5:1 || Near-stoichiometric; good combustion efficiency but higher emissions.
|-
| 16:1 || Balanced performance; good power output and efficiency.
|-
| 18:1 || Lean burn; improved fuel economy but potential for higher NOx emissions.
|}

Diesel is a complicated compound, but it’s commonly approximated as a hydrocarbon like C12H23 (or C12H26). Air is about 21% O2 and 79% N2. Without nitrogen, the stoichiometric ratio is about

C12H23+17.75O2→12CO2+11.5H2O

and by including nitrogen, we get

C12H23+17.75( O2 + 3.76 N2 )→12CO2+11.5H2O + 66.74 N2.

The molar masses
* Fuel: 167 g/mol
* Air: 137.28 g/mol
And the total air needed is 17.72 x 137.28 = 2436 g, which gives air-to-fuel-ratio to

<math>
AFR = \frac{2436}{167} = 14.6:1
</math>

=== MB data ===

Motor: OM648.

Mercedes Benz W211 (2003)
* Engine displacement: 3222 cm3 = 0.003222 m3
* Bore x Stroke: 88.0 x 88.4 mm3
* Compression Ratio: 18.0

Bore x stroke gives V = 6xπ(8.8/2)2 x 8.84 cm3 = 3225.95cm3, which is rather close.

<math>
p_1 V_1^\gamma = \text{constant}_1
</math>

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