Sensible Heat Lost by a Cooling Flue Gas: 23-Chem-A1 Process Balances and Chemical Thermodynamics, May 2017, Q3

Question 3: Sensible Heat Lost by a Cooling Flue Gas (Part A)

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Given. Mole fractions CO₂ 0.12, H₂O 0.13, O₂ 0.02, N₂ 0.73; heat-capacity coefficients $C_p^\circ = a + bT + cT^2$ (kJ/kmol·K); cool 10 kmol from $T_1=650\ \text{K}$ to $T_2=300\ \text{K}$.

Component	$a$	$b\times10^{3}$	$c\times10^{6}$
CO₂	21.35	64.27	−41.01
H₂O	32.49	0.08	13.21
O₂	26.01	11.76	−2.35
N₂	29.60	−5.15	13.19

Find. The sensible heat released, $Q$, on cooling the 10 kmol from 650 K to 300 K.

Approach. Mole-average the polynomial coefficients, integrate $C_p$ over the temperature interval for one kmol, then scale to 10 kmol.

Mixture heat-capacity coefficients. For an ideal-gas mixture the extensive $C_p$ is additive, so the coefficients are mole-weighted sums: $$a_m = \sum y_i a_i = 28.91,\quad b_m = 4.199\times10^{-3},\quad c_m = 6.378\times10^{-6}\ \ (\text{kJ/kmol}\cdot\text{K})$$
Integrate the heat capacity. The sensible heat per kmol between $T_1$ and $T_2$ is $$q = a_m\,\Delta T + \frac{b_m}{2}\left(T_2^2 - T_1^2\right) + \frac{c_m}{3}\left(T_2^3 - T_1^3\right)$$ $$q = 28.91(350) + \tfrac{4.199\times10^{-3}}{2}(3.325\times10^{5}) + \tfrac{6.378\times10^{-6}}{3}(2.4763\times10^{8}) = 10120 + 698 + 526 = 11344\ \text{kJ/kmol}$$
Scale to 10 kmol. The magnitude of heat lost on cooling is $$Q = 10\,q = \boxed{1.134\times10^{5}\ \text{kJ} \approx 113.4\ \text{MJ}}$$

Quantity	Value
Mixture $a_m,\ b_m,\ c_m$	28.91, 4.199×10⁻³, 6.378×10⁻⁶
Heat per kmol (650→300 K)	11 344 kJ/kmol
Heat lost by 10 kmol	113.4 MJ

Concept & Theory: Temperature-Dependent $C_p$ and Stack Losses

Topic. Sensible-heat accounting for a cooling gas mixture with polynomial heat capacities.

Key relations. $Q = n\int_{T_1}^{T_2} C_p\,dT$; for the mixture $C_{p,m} = \sum y_i C_{p,i}$, term-by-term integration of $a+bT+cT^2$.

Why this works. Enthalpy is a state function and, for an ideal-gas mixture, additive with no interaction terms, so the mole-weighted coefficients integrate exactly like a single species.

Common pitfall. Using a constant $C_p$ (under-predicts the loss, since polyatomic $C_p$ climbs with $T$ as vibrational modes activate) or mixing up the $10^{3}$/$10^{6}$ scaling on $b$ and $c$.

Source. Smith, Van Ness & Abbott, Introduction to Chemical Engineering Thermodynamics, 8th ed., §4.1 (ideal-gas heat capacities); Felder & Rousseau, Ch. 8 (sensible-heat energy balances).

Question 3 of 6: Sensible Heat Lost by a Cooling Flue Gas

Question 3: Sensible Heat Lost by a Cooling Flue Gas (Part A)