Lecture 11

1. Introduction & Context

To solve for the three operating variables of the Blast Furnace (Coke Rate, Blast Rate, Top Gas Composition), we need three characteristic equations.

• Equation 1: Overall Oxygen Balance (Derived in Lecture 10).

• Equation 2: Bottom Segment Oxygen Balance (Derived in Lecture 10).

• Equation 3: Enthalpy Balance (Subject of this lecture).

Why “Bottom Segment” Enthalpy Balance?

We avoid performing an enthalpy balance over the entire furnace because the Top Gas Temperature ($T_{top}$) is an unknown process variable.

• If we used the whole furnace, we would introduce a 4th unknown ($T_{top}$).

• By focusing on the Bottom Segment (bounded by the Thermal Reserve Zone at the top and the Hearth at the bottom), all boundary temperatures are fixed:

  • Upper Boundary: Gas and Solids are at Thermal Equilibrium $\approx 1200\text{ K}$.

  • Lower Boundary: Liquid products leave at $\approx 1800\text{ K}$.

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2. Control Volume & Boundary Conditions

Control Volume: The region from the Thermal Reserve Zone (TRZ) down to the Hearth.

Basis of Calculation: 1 mole of Fe product.

Inputs (Entering the Control Volume)

Material	Composition / Species	Moles	Temperature
Solids (from Top)	Wustite ($F{e}_{0.947}O$)	$1.06$	$1200\text{ K}$
	Carbon (Coke)	$N_{IC}$	$1200\text{ K}$
Gases (from Bottom)	Blast Oxygen ($O_2$)	$\frac{1}{2}{N}_{OB}$	$T_{blast}$ (Variable)
	Blast Nitrogen ($N_2$)	$\frac{1}{2}{N}_{N_2}^{blast}$	$T_{blast}$

Outputs (Leaving the Control Volume)

Material	Composition / Species	Moles	Temperature
Liquids (to Hearth)	Liquid Iron ($F{e}_{(l)}$)	$1.0$	$1800\text{ K}$
	Dissolved Carbon ($\underline{C}$)	$(C/Fe{)}_m$	$1800\text{ K}$
Gases (to Top)	Carbon Monoxide ($CO$)	$n_{CO}^{gas}$	$1200\text{ K}$
	Carbon Dioxide ($C{O}_2$)	$n_{C{O}_2}^{gas}$	$1200\text{ K}$
	Nitrogen ($N_2$)	$\frac{1}{2}{N}_{N_2}^{blast}$	$1200\text{ K}$

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3. Formulation of the Heat Balance

At steady state: Total Enthalpy Input = Total Enthalpy Output

$\sum (n_i{H}_i{)}_{in}=\sum (n_i{H}_i{)}_{out}$

3.1. The Raw Equation

Left Hand Side (Heat Input):

$\begin{array}{rl}\text{Input}& =\underset{\text{Wustite}}{\underbrace{1.06\cdot H^{\circ }(F{e}_{0.947}O,1200)}}+\underset{\text{Coke Carbon}}{\underbrace{N_{IC}\cdot H^{\circ }(C,1200)}}\\ & +\underset{\text{Blast Oxygen}}{\underbrace{\frac{1}{2}{N}_{OB}\cdot H^{\circ }(O_2,T_{blast})}}+\underset{\text{Blast Nitrogen}}{\underbrace{\frac{1}{2}{N}_{N_2}\cdot H^{\circ }(N_2,T_{blast})}}\end{array}$

Right Hand Side (Heat Output):

$\begin{array}{rl}\text{Output}& =\underset{\text{Liquid Iron}}{\underbrace{1.0\cdot H(F{e}_{(l)},1800)}}+\underset{\text{Dissolved Carbon}}{\underbrace{(C/Fe{)}_m\cdot H(\underline{C},1800)}}\\ & +\underset{\text{Gas CO}}{\underbrace{n_{CO}\cdot H^{\circ }(CO,1200)}}+\underset{\text{Gas }C{O}_2}{\underbrace{n_{C{O}_2}\cdot H^{\circ }(C{O}_2,1200)}}\\ & +\underset{\text{Gas Nitrogen}}{\underbrace{\frac{1}{2}{N}_{N_2}\cdot H^{\circ }(N_2,1200)}}\end{array}$

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4. Simplification & Grouping

To make this equation solvable and elegant, we perform algebraic manipulations to group terms into meaningful process parameters.

Step 1: Handling Carbon Terms ($N_{IC}$)

The input carbon ($N_{IC}$) splits into Active Carbon ($N_{AC}$) which becomes gas, and Passive Carbon ($(C/Fe{)}_m$) which dissolves in iron.

$N_{IC}=N_{AC}+(C/Fe{)}_m$

• The passive carbon term on the LHS is grouped with the dissolved carbon term on the RHS.

• The active carbon ($N_{AC}$) is grouped with the gas terms ($CO,C{O}_2$).

Step 2: Simplifying Gas Composition

Using the relationships derived in Lecture 1:

• $n_{CO}=N_{AC}\cdot (2-O/C_{gas})$

• $n_{C{O}_2}=N_{AC}\cdot (O/C_{gas}-1)$

Step 3: The “Blast Enthalpy” ($E_B$) Transformation

This is the most critical derivation step. We aim to group all Blast-related terms ($O_2$ and $N_2$) into a single parameter $E_B$.

1. Nitrogen Terms: Group Input ($T_B$) and Output ($1200$) Nitrogen terms.

   $\text{Net }{N}_2\text{ Heat}=\frac{1}{2}{N}_{N_2}[H^{\circ }(N_2,T_B)-H^{\circ }(N_2,1200)]$

2. Oxygen Terms: The $O_2$ enters at $T_B$ but “leaves” as $CO/C{O}_2$. To mathematically group it like Nitrogen, we add and subtract a dummy term: $\frac{1}{2}{N}_{OB}{H}^{\circ }(O_2,1200)$.

   This allows us to form a similar difference term for Oxygen:

   $\text{Net }{O}_2\text{ Sensible Heat}=\frac{1}{2}{N}_{OB}[H^{\circ }(O_2,T_B)-H^{\circ }(O_2,1200)]$

3. Defining $E_B$:

   We define the Enthalpy of Blast ($E_B$) per mole of Oxygen ($N_{OB}$):

   $E_B=\frac{1}{2}\left[(H_{O_2,T_B}^{\circ }-H_{O_2,1200}^{\circ })+\frac{0.79}{0.21}(H_{N_2,T_B}^{\circ }-H_{N_2,1200}^{\circ })\right]$

   • Note: The factor $\frac{0.79}{0.21}$ accounts for the ratio of $N_2$ to $O_2$ in air.

Physical Interpretation of $E_B$

• If $T_{blast}=1200\text{ K}$: $E_B=0$. The blast brings no “extra” heat relative to the control volume boundary temperature.

• If $T_{blast}>1200\text{ K}$: $E_B>0$. The blast acts as a Heat Supply.

• If $T_{blast}<1200\text{ K}$: $E_B<0$. The blast creates a Heat Demand (needs heating).

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5. Intermediate Result: The Transposed Equation

After rearranging terms, grouping the blast enthalpy, and substituting carbon fractions, the equation takes the form:

$\begin{array}{r}\underset{\text{Heat Supply from Blast}}{\underbrace{E_B\cdot N_{OB}}}\&+\text{ [Remaining Reaction \& Solid Terms] }\\ \&=\text{ [Liquid Output Terms] }\end{array}$

• The derivation essentially separates terms into Heat Supply (Combustion + Hot Blast) and Heat Demand (Reduction of FeO + Melting/Heating Products).

• Instructor’s Note: The final “Supply = Demand” form ($D_E=S_E$) will be finalized in the next lecture.

6. Summary of Key Concepts

1. The W Point: The intersection of the Overall and Bottom Segment operating lines on the Rist Diagram occurs at $W(1.3,1.06)$. This point is a characteristic constant for a stable Thermal Reserve Zone.

2. Adiabatic Assumption: The derivation assumes no heat loss through walls (Adiabatic).

3. Reference State: The choice of 1200 K as the reference for gas separation simplifies the math significantly, avoiding the unknown Top Gas Temperature.