Adiabatic-Mixing Temperature of a Cleaning Solution: 23-Chem-A1 Process Balances and Chemical Thermodynamics, December 2019, Q6

Question B3: Adiabatic-Mixing Temperature of a Cleaning Solution (Part B — equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Equal masses of acetone and dichloromethane, both initially at 298 K, are mixed adiabatically at 1 bar with negligible stirring work.

Property	Value
$C_p$ acetone @ 1 bar, 298 K	2.173 kJ/kg·K
$C_p$ dichloromethane @ 1 bar, 298 K	1.193 kJ/kg·K
Heat of mixing (equal mass) @ 293 / 298 / 303 K	12.468 / 12.380 / 12.292 kJ/kg

Find. the temperature of the cleaning solution leaving the adiabatic mixer.

Figure B3 — Two equal-mass feeds at 298 K enter an adiabatic mixer; the exothermic heat of mixing is retained (Q=0) and warms the product solution above the feed temperature.

Approach. Conserve enthalpy across the adiabatic mixer: the exothermic heat of mixing at the 298-K inlet is absorbed as sensible heat by the product, whose mass-average heat capacity sets the temperature rise.

Heat capacity of the solution. For an equal-mass blend the mass-average is $$C_{p,soln}=\tfrac12(2.173+1.193)=1.683\ \text{kJ/kg}\cdot\text{K}.$$
Adiabatic enthalpy balance. At constant pressure with $Q=0$ and no work the total enthalpy is conserved. Along the path “mix at 298 K, then warm the product,” the exothermic heat of mixing released at 298 K (12.380 kJ per kg of solution) is entirely absorbed as sensible heat: $$0=\Delta H_{mix}(298)+C_{p,soln}(T-298)\;\Rightarrow\; T-298=\frac{12.380}{1.683}=7.36\ \text{K}.$$
Final temperature. $$T=298+7.36=\boxed{305.4\ \text{K}\ (32.2\ ^\circ\text{C})}.$$ The 293- and 303-K heat-of-mixing values differ from the 298-K value by only ~1.4% over 10 K, so using the inlet (298 K) value is accurate; re-evaluating $\Delta H_{mix}$ at the 305-K outlet shifts $T$ by under 0.1 K.

Quantity	Result
Solution heat capacity	1.683 kJ/kg·K
Temperature rise	7.36 K
Cleaning-solution temperature	305.4 K (32.2 °C)

Concept & Theory

Topic. Adiabatic mixing with an exothermic heat of solution.

Key relations. Constant-$P$ adiabatic energy balance $\Delta H_{total}=0$; heat of mixing $\Delta H_{mix}(T)$; mass-average $C_{p,soln}=\sum w_i C_{p,i}$.

Why this works. Mixing acetone and dichloromethane is exothermic (negative deviation from ideality: the two form a weak hydrogen-bonded complex), so the heat that would otherwise be released to the surroundings instead stays in the adiabatic system and raises its temperature. Equating the heat of mixing at the inlet temperature to the sensible-heat rise of the product gives the outlet temperature in one step.

Common pitfall. Sign errors: an exothermic $\Delta H_{mix}$ raises $T$; treating it as an ideal (zero) mix or flipping the sign gives no rise or a spurious cooling. Also use a mass-average $C_p$, since the data are per unit mass.

Source. Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics §12.4 (heat effects of mixing; enthalpy–concentration and adiabatic-mixing balances).

Question 6 of 6: Adiabatic-Mixing Temperature of a Cleaning Solution

Part A — Process Mass & Energy Balances

Question B3: Adiabatic-Mixing Temperature of a Cleaning Solution (Part B — equal value)