The Rd–C Diagram for Blast Furnace
Why can increasing direct reduction first lower, then raise, the carbon requirement? Explore the competing demands for reducing gas and heat in a simplified blast-furnace teaching model.
What this diagram tells you
The horizontal axis is the modeled direct-reduction share Rd. The vertical axis is carbon in kg C per tonne of hot metal (tHM). The two main curves show a gas-based requirement and a heat-based requirement. Their comparison illustrates a trade-off; it is not a prediction of an operating furnace's coke rate.
Carbon requirement versus direct reduction
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Click the chart to move the Rd probe. With chart keyboard focus, ← / → changes Rd by 0.01. Exact values are available in the table.
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Conditional on the selected assumptions; not a plant operating optimum.
Read the selected point
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| Term | kg C/tHM | Meaning in this model |
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A plain-language guide
Carbon does more than one job in a blast furnace. It helps remove oxygen from the iron-bearing burden, produces CO for indirect reduction, supplies heat, and dissolves into hot metal. Changing the share of direct reduction changes these demands in opposite directions.
- Choose a heat demand and gas-utilization scenario. These inputs define two possible requirements for the same carbon supply.
- Read the descending gas curve. With more direct reduction, less modeled FeO-stage reduction is assigned to CO. The source gas requirement therefore falls with Rd.
- Read the ascending heat curve. Direct reduction consumes carbon and increases the source model's heating burden. The heat-based carbon requirement therefore rises with Rd.
- Follow the higher requirement. The dashed green curve takes the larger of the source gas requirement, the source heat requirement, and a carbon sink floor. Where one constraint relaxes while another tightens, this simplified bound can have a low point.
Why take the maximum instead of adding the curves?
Carbon that contributes to heat can also form the CO that reduces iron oxide. The gas and heat curves are two constraints on a shared supply. Adding their totals would count some carbon duties twice. Taking their maximum is a teaching construction used here to compare the source curves; it does not close a full furnace carbon, oxygen, hydrogen and energy balance.
A worked example that follows your inputs
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What Rd means here
The source notebook multiplies Rd by one oxygen atom per iron atom, using the FeO → Fe stage. In this web model, Rd is therefore the share of that modeled stage assigned to the net reaction FeO + C → Fe + CO. The remainder is assigned to indirect reduction by CO. This net direct-reduction reaction can represent reduction coupled with CO₂ gasification; it need not mean solid carbon touching every oxide particle. It also does not mean the industrial direct-reduced iron (DRI) process.
A conventional whole-furnace direct-reduction degree may use a different oxygen balance, including reduction from hematite and magnetite. The notebook does not explicitly balance those upstream stages. Do not treat this Rd as interchangeable with a plant-reported value without reconciling their definitions.
How this differs from the other diagrams
Ellingham compares standard Gibbs energies of reactions. Baur–Glässner maps equilibrium stability against temperature and gas composition. This Rd–C diagram compares simplified gas and heat requirements against a direct-reduction share. It adds a teaching-level engineering correlation, but it is not a spatial furnace model. A Rist diagram uses different oxidation and material-balance coordinates; this is an Rd–C plot, not a Rist operating diagram.
Equations, carbon accounting and units
a = (12 / 56) × 951.3 = 203.85 kg C/tHM
Cd = a Rd
Cgas = a (1 − Rd) / η + 48.7 + 5.87
H = vair cair Tblast, vair = (22.4 / 24) / 0.21
Cheat,raw = 1000 × [(1000 Q − 5073) + (5073 + 0.215 H) Rd] / (9791 + H)
Cthermal = Cd + Cheat,raw + 48.7
Cfloor = Cd + 48.7 + 5.87
Cbound = max(Cgas, Cthermal, Cfloor)
Q is entered in GJ/tHM; Tblast is in °C, following the original correlation. Dimensional interpretation required for the source formula: 5073 kJ/kg hot metal, 9791 kJ/kg C and 0.215 kg C/kg hot metal. These units are inferred for consistency; the notebook does not document the coefficients' derivation. H is kJ/kg C, with air sensible heat referenced to 0 °C. Before the outer factor 1000, the heat expression is kg C/kg hot metal. Multiplying by 1000 kg hot metal/tHM converts it to kg C/tHM. The floor is a web-model accounting guard that retains carbon consumed by direct reduction and the stated fixed sinks at high Rd; it was not drawn in the original notebook.
The source gas line includes dissolved carbon and other reduction carbon; its heat line includes dissolved carbon but not the 5.87 kg C/tHM other-reduction term. The web model preserves both source lines rather than silently making them identical. Their maximum is consequently an illustrative bound, not a rigorously reconciled optimization model.
In equilibrium mode: K = exp(−ΔG° / RT), n = 1 + 1/K and η = 1/n = K/(1 + K), for FeO + CO → Fe + CO₂.
Fixed source assumptions and what is excluded
| Quantity | Value | Interpretation / status |
|---|---|---|
| Hot metal basis | 1000 kg/tHM | 48.7 kg dissolved carbon + 951.3 kg nominal Fe; the basis leaves no separate impurity mass |
| Other-reduction carbon | 5.87 kg C/tHM | Fixed source assumption, without an explicit separate species balance |
| Molar masses | C = 12; Fe = 56 g/mol | Rounded values used in the notebook |
| Air oxygen fraction | 0.21 | Dry-air approximation |
| Gas molar volume | 22.4 Nm³/kmol | Source standard-volume convention; used to estimate air per kg C burned to CO |
| Air sensible-heat coefficient | 1.436 kJ/(Nm³·°C) | Fixed source coefficient, with sensible heat referenced to 0 °C; no temperature-dependent integration |
| Heat-correlation constants | 5073; 9791; 0.215 | Inherited source coefficients. The notebook gives no documented derivation or calibration. The heat factor 0.215 differs from a/1000 = 0.20385 used by its direct-carbon line; no independently verified plant validity is claimed. |
Excluded: burden chemistry and gangue, slag amount, moisture, ore quality, oxygen enrichment, injected fuels and hydrogen, separate coke / coal qualities, heat losses calculated from furnace geometry, kinetics, gas flow, shaft efficiency, permeability and coke's structural role. The upstream energy used to preheat the blast is outside this correlation.
All carbon outputs are elemental carbon on a tHM basis. They are neither total coke mass nor total fossil-fuel consumption. This model does not decide how low a real coke rate may safely go.
Equilibrium mode and historical data cautions
The optional equilibrium mode reproduces the original course thermodynamic functions for pure Fe, nominal stoichiometric FeO, CO and CO₂. It treats the reactive gas pair as ideal and uses unit activities for condensed phases. The estimate is a reference for an Fe/FeO equilibrium pair, not a fitted furnace gas-utilization efficiency.
The stable range of real non-stoichiometric wüstite differs from a pure stoichiometric approximation. The original 800 K (526.85 °C) example lies below the commonly cited ≈570 °C wüstite stability limit; it remains available only to reproduce the historical teaching curve, with a warning. The selectable source range is 800–1600 K; the web model does not extrapolate outside it.
The active source-data warnings appear below the result table.
Sources and reproducibility
Based on Kejiang Li's course notebook “The Rd-C Diagram for Blast Furnace”, updated 9 March 2024, and its accompanying ThermodynamicsData.py. The gas and heat lines are reproduced exactly; the combined envelope and carbon sink floor are explicitly identified web-model additions.
Reference overlays reproduce the notebook's three temperature and four heat-demand cases. CSV exports include every selected input, source coefficients, sampled curves, the probe and the conditional minimum. SVG exports retain the legend and assumptions.