Classical Köhler Theory vs κ-Köhler Theory in A Study Accommodating Black Carbon (BC) heating

Excellent — you're using the classical Köhler theory to compute the equilibrium saturation ratio (Seq), and you’re asking how this compares to the κ-Köhler theory approach. Let’s explain the difference clearly and provide the equations.

Note: As in Nenes (2002), there are 4 scenarios how the presence of BC affect the formed DSD:

Scenario 1 (NoHeat scenario): just as the basic assumption in cloud parcel model - there's only latent heat release upon condensation.

dT/dt=adiabatic cooling+latent heating

Scenario 2 (Drop scenario): There's additional heat (reflected by ga value) caused by light absorption of BC in addition to the basic latent heat; this additional heat influence only the Seq of the water droplet.

dT/dt=adiabatic cooling+latent heating (No change); but Seq change

Scenario 3 (External Scenario); The BC is not part of the droplet, i.e. externally mixed particle with the hygroscopic or water absorbing particles, so the BC doesn't heat the droplet but heat the parcel that influence the net dT/dt of the parcel (adiabatic cooling + droplet condensation latent heat + additional external BC heat referred as $Q_{parcel} / (ρ_{air} c_{p})$ .

dT/dt=adiabatic cooling+latent heating+BC parcel heating (gas)

Scenario 4 (Internal Scenario): The BC is part of the droplet and has effects on both the droplet and gas parcel - accommodate both the drop scenario effect (scenario 2) and external scenario effect (scenario 3) eventhough the BC heat the parcel from with the droplet - both the Seq and dT/dt change

⚙️ Estimate of Parcel Heating Power due to BC (Q0_parcel)

$Q_{parcel} = f_{BC} \cdot Q_{0}^{parcel} [W/m³]$

Where:

$f_{\text{BC}}$ : fraction of BC relative to some reference.
$Q_0^{\text{parcel}}$ : power absorbed per cubic meter of air when $f_{\text{BC}} = 1$

🌫️ 1. Classical Köhler Theory (What You Use)

This theory calculates the equilibrium water vapor saturation ratio over a droplet as a combination of:

Curvature effect (Kelvin term): smaller droplets have higher surface curvature, which increases vapor pressure.
Solute effect (Raoult’s law): dissolved salt lowers the vapor pressure.

Equation

S_\text{eq}(D) = \exp\left( \frac{A}{D} - \frac{B}{D^3 - D_\text{dry}^3}

where:

$D$ : droplet diameter (wet)
$D_\text{dry}$ : dry particle diameter
$A = \frac{2\sigma M_w}{R T \rho_w}$ : curvature effect
$B = \frac{6 \nu M_s}{\pi \rho_w}$ : solute effect (with van't Hoff factor $\nu$ )

Your code implements this version, with optional enhancement from g_a to model heating effects (black carbon), i.e.:

S_\text{eq}(D) = \exp\left( \frac{A (1 + g_a)}{D} - \frac{B}{D^3 - D_\text{dry}^3} \right)

This is the traditional Köhler equation and works well when you know the chemical composition (e.g., ammonium sulfate), since $\nu$ , $M_s$ are explicitly defined.

🌧️ 2. κ-Köhler Theory (Simplified, Empirical)

Petters & Kreidenweis (2007) proposed this as a more compact and generalizable formulation. It simplifies solute effects into a single parameter $\kappa$ , which captures the particle's hygroscopicity.

Equation

S_\text{eq}(D) = \left(1 + \frac{\kappa D_\text{dry}^3}{D^3 - D_\text{dry}^3}\right) \cdot \exp\left(\frac{A}{D}\right)

where:

$\kappa$ : empirical hygroscopicity parameter
- ~0.6 for ammonium sulfate
- ~0.1 for organics
- ~0.0 for black carbon

The Raoult (solute) term is simplified using $\kappa$ , which bundles all chemistry (valence, molecular weight, density) into one parameter.

✅ When to Use Which

Criterion	Classical Köhler	κ-Köhler
Needs chemical info?	Yes (e.g., $\nu$ , $M_s$ )	No, only κ
Physics transparency	High	Lower (semi-empirical)
Simplicity	More complex	Simpler, more general
Aerosol diversity	Limited (needs speciation)	Good for mixed/incomplete data
Black carbon heating	Modifiable (as in your code)	Rarely included directly

✍️ Example Values

Species	κ
Ammonium sulfate	0.6
Sea salt	~1.2
Organics (low O:C)	0.1–0.2
Black carbon	~0.0

🧠 Final Note

Your current formulation is better for explicit microphysics with known chemistry and when you need to model per-mode activation (as in Nenes et al. 2001/2002).
κ-Köhler is excellent for speed and bulk CCN modeling, especially when the chemical mix is unknown or simplified.

#######################

Great. Let’s convert your classical Köhler equation into the κ-Köhler form in your MATLAB code, and then allow a switch between the two formulations depending on the input.

🔁 Modified Code: Add κ-Köhler Option

First, introduce a parameter in your structure p to select the Köhler method:

p.scenario = 'Drop';     % Options: 'Drop', 'Internal', 'External', 'NoHeat', etc.
p.f_BC = 0.05;           % Fractional BC content for heating calculation

valid_scenarios = {'Drop', 'Internal', 'External', 'NoHeat'};
if ~ismember(p.scenario, valid_scenarios)
    error('Unknown scenario: %s. Valid options: %s', p.scenario, strjoin(valid_scenarios, ', '));
end
% p.kohler_type = 'classic';   % or 'kappa'
% p.kappa = 0.6;               % e.g., for ammonium sulfate

Then modify your Köhler block like this:


% === Köhler parameters ===
sigma = 0.072;           % Surface tension [N/m]
Mw = 0.018;              % Molar mass of water [kg/mol]
R = 8.314;               % Universal gas constant [J/mol/K]
rho_w = 1000;            % Water density [kg/m³]
A = (2 * sigma * Mw) / (R * T * rho_w);

Dp_wet = max(2 * r, 1e-9);          % Wet droplet diameter [m]
Dp_dry = max(p.Dp, 1e-9);          % Dry particle diameter [m]

% === Optional g_a: droplet BC heating enhancement ===
ka = 0.024;                % Thermal conductivity of air [W/m/K]
L = 2.5e6;                 % Latent heat of vaporization [J/kg]
Q0 = 1e-12;                % Power per droplet at f_BC = 1 [W]
g_a = zeros(size(r));
if strcmp(p.scenario, 'Drop') || strcmp(p.scenario, 'Internal')
    Q_drop = p.f_BC * Q0;
    Hv = L / T;
    g_a = (rho_w * Q_drop * Hv) ./ (8 * pi * ka * T .* r * L);
end

% === Köhler Equation (classic or kappa) ===
switch p.kohler_type
    case 'classic'
        nu = 2;                  % van’t Hoff factor for (NH₄)₂SO₄
        Ms = 0.132;              % Molar mass of (NH₄)₂SO₄ [kg/mol]
        B = (6 * nu * Ms) / (pi * rho_w);
        volume_diff = Dp_wet.^3 - Dp_dry.^3;
        volume_diff(volume_diff <= 0) = 1e-21;
        Seq = exp(min(max((A * (1 + g_a)) ./ Dp_wet - B ./ volume_diff, -5), 5));
        
    case 'kappa'
        kappa = p.kappa;         % e.g., 0.6 for ammonium sulfate
        volume_diff = Dp_wet.^3 - Dp_dry.^3;
        volume_diff(volume_diff <= 0) = 1e-21;
        Raoult_term = (1 + (kappa * Dp_dry.^3) ./ volume_diff);
        Seq = Raoult_term .* exp(min(max((A * (1 + g_a)) ./ Dp_wet, -5), 5));
        
    otherwise
        error('Unknown Köhler formulation: %s', p.kohler_type);
end

✅ Result

This block now supports both:

Classical Köhler theory (with explicit chemistry)
κ-Köhler theory (more general, empirical)

You can control the switch via:

p.kohler_type = 'kappa';  % or 'classic'
p.kappa = 0.6;

#################

Great — let's walk through how to compare the two Köhler formulations visually in MATLAB: the classical Köhler and the κ-Köhler theory, by plotting the equilibrium saturation ratio $S_\text{eq}$ versus droplet diameter $D_p$ .

🔬 Objective

We’ll generate a plot of:

$S_{eq} (D_{p})$ :
- Classical Köhler equation
- κ-Köhler equation

📈 Step-by-Step MATLAB Script

Here’s a self-contained script you can run to visualize both curves for comparison:

% === Parameters ===
T = 298;                        % Temperature [K]
sigma = 0.072;                  % Surface tension [N/m]
Mw = 0.018;                     % Molar mass of water [kg/mol]
R = 8.314;                      % Universal gas constant [J/mol/K]
rho_w = 1000;                   % Water density [kg/m³]
Dp_dry = 0.05e-6;               % Dry particle diameter [m] (50 nm)

% === Classical Köhler parameters ===
nu = 2;                         % van’t Hoff factor for (NH4)2SO4
Ms = 0.132;                     % Molar mass [kg/mol]
B = (6 * nu * Ms) / (pi * rho_w);

% === κ-Köhler parameters ===
kappa = 0.6;

% === Common A parameter ===
A = (2 * sigma * Mw) / (R * T * rho_w);

% === Droplet radius range ===
r = logspace(log10(Dp_dry/2), log10(2e-6), 500);   % [m]
Dp_wet = 2 * r;

% === Common terms ===
volume_diff = Dp_wet.^3 - Dp_dry.^3;
volume_diff(volume_diff <= 0) = 1e-21;

% === Classical Köhler Equation ===
Seq_classic = exp((A ./ Dp_wet) - (B ./ volume_diff));

% === κ-Köhler Equation ===
Raoult_term = 1 + (kappa * Dp_dry^3) ./ volume_diff;
Seq_kappa = Raoult_term .* exp(A ./ Dp_wet);

% === Plotting ===
figure;
semilogx(Dp_wet * 1e9, Seq_classic, 'b-', 'LineWidth', 2); hold on;
semilogx(Dp_wet * 1e9, Seq_kappa, 'r--', 'LineWidth', 2);
xlabel('Wet diameter D_p (nm)');
ylabel('Equilibrium Saturation Ratio S_{eq}');
legend('Classical Köhler', '\kappa-Köhler');
title(['Köhler Theory Comparison, D_p_{dry} = ' num2str(Dp_dry*1e9, '%.0f') ' nm']);
grid on;

📌 Notes

You can change Dp_dry to compare activation curves for different dry particle sizes.
This assumes no heating, i.e., $g_{a} = 0$
You can add a third curve if you want to include black carbon heating enhancement by modifying the exponent:
$\exp\left( \frac{A (1 + g_a)}{D_p} \right)$

So, can k -kohler theory be used to study BC heating effect?

❌ **κ-Köhler theory cannot directly model BC heating effects.**

Why?

The κ-Köhler theory is a thermodynamic equilibrium framework used to describe hygroscopic growth and activation of aerosol particles. It accounts for:

Water activity reduction via solute (Raoult's law) → parameterized by κ
Curvature effect (Kelvin term)

It's useful to compute critical supersaturation (Sc) and critical diameter (Dc) for droplet activation.

But BC Heating is a non-equilibrium, radiative-thermal effect, involving:

Absorption of solar radiation by black carbon (BC),
Temperature increase inside the droplet,
Modification of droplet growth rate and possibly feedback to parcel temperature or supersaturation,
Time-dependent dynamics, not captured by κ-Köhler’s steady-state formulation.

How they relate (indirectly):

You can combine κ-Köhler theory with a dynamic cloud parcel model that includes:

Radiative heating (via black carbon absorption),
Droplet temperature evolution,
Enhanced vapor pressure or modified growth rates.

In that setup:

κ still governs how hygroscopic the particle is (how much water it attracts),
But BC heating modifies growth kinetics, not the equilibrium activation condition.

Summary Table:

Aspect	κ-Köhler Theory	BC Heating Effect
Type of model	Equilibrium thermodynamics	Time-dependent thermal dynamics
Affects	Critical supersaturation	Droplet temperature, growth rate
Parameters	κ (hygroscopicity)	Absorptivity, radiation, `ga`, etc.
Handles heating?	❌ No	✅ If coupled to heat/radiation model
Suitable for BC studies?	❌ Not standalone	✅ When extended with heat physics

Best Practice:

Use κ-Köhler for initial activation predictions, then extend with:

Dynamic parcel model including heat terms,
BC mass and location (internal/external mixing),
Optical properties of BC (absorption cross section, etc.).

Check the next posts to see how possible BC heating effects are studied while using the k-Kohler theory instead of the classic Kohler theory!!

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