Biohygrothermal Models

Biohygrothermal models couple mold growth predictions with dynamic heat and moisture transport simulations in building assemblies. These models predict mold development under transient boundary conditions by solving simultaneous differential equations for temperature, moisture, and biological activity.

VTT Mold Growth Model

Model Structure

The VTT model (Technical Research Centre of Finland) integrates biological growth kinetics with material hygrothermal properties.

Mold Index Evolution:

dM/dt = (1/7) × k₁ × k₂ × exp(-0.68 ln(RHcrit) - 13.9 ln(RHcrit)/T + 0.14 W - 0.33 ln(t₁))

Where:

• M = mold index (0-6 scale)
• t = time (weeks)
• k₁ = intensity factor based on material sensitivity
• k₂ = surface quality factor
• RHcrit = critical relative humidity
• T = temperature (°C)
• W = wood species factor
• t₁ = delay time for growth initiation

Critical Conditions

Material Sensitivity Classes:

Decline Phase

When conditions become unfavorable (RH < RHcrit):

dM/dt = -0.032 (for M ≥ 3)
dM/dt = -0.016 (for M ≥ 1 and M < 3)

Mold index decreases but never returns to zero, representing permanent colonization.

ESP-r Mold Model

Integration Approach

ESP-r (Environmental Systems Performance - research) couples mold prediction with whole-building energy and moisture simulation.

Simultaneous Solution:

1. Heat transfer equation:

ρc ∂T/∂t = ∇(λ∇T) + Lv ∂u/∂t

2. Moisture transport equation:

ξ ∂φ/∂t = ∇(Dφ∇φ + δp∇(φpsat))

3. Mold growth rate:

dM/dt = f(T, φ, substrate, M)

Where:

• ρ = material density (kg/m³)
• c = specific heat capacity (J/kg·K)
• λ = thermal conductivity (W/m·K)
• Lv = latent heat of vaporization (2.5×10⁶ J/kg)
• u = moisture content (kg/m³)
• ξ = moisture capacity (kg/m³)
• φ = relative humidity (-)
• Dφ = liquid diffusivity (m²/s)
• δp = vapor permeability (kg/m·s·Pa)
• psat = saturation vapor pressure (Pa)

Substrate Parameterization

Material-specific growth parameters integrated into building component database:

• Sorption isotherm coefficients
• Critical RH thresholds
• Nutrient availability indices
• Surface pH effects
• Biocide presence factors

Moisture Transport Coupling

Capillary-Vapor Transfer

Combined Moisture Flux:

g = -δp∇(φpsat) - Dw∇u

Where:

• g = total moisture flux (kg/m²·s)
• δp = vapor permeability (kg/m·s·Pa)
• Dw = liquid water diffusivity (m²/s)

Sorption Hysteresis

Absorption vs Desorption:

Materials exhibit different moisture storage curves depending on wetting history:

u_abs = A₁(1 - exp(-B₁φ^C₁))
u_des = A₂(1 - exp(-B₂φ^C₂))

Hysteresis affects mold prediction because spore germination occurs preferentially during absorption phases.

Surface Exchange

Boundary Condition:

g_surf = βx(pv,air - φpsat,surf)

Where:

• βx = surface moisture transfer coefficient (kg/m²·s·Pa)
• pv,air = air vapor pressure (Pa)
• φpsat,surf = surface vapor pressure (Pa)

Typical βx values: 2×10⁻⁸ to 8×10⁻⁸ kg/m²·s·Pa

Spore Germination Modeling

Germination Probability

Time-Dependent Activation:

P(t) = 1 - exp(-k_germ × t^n)

Where:

• P(t) = probability of germination at time t
• k_germ = germination rate constant (1/day)
• n = shape parameter (typically 1.5-2.5)

Environmental Dependencies

Temperature Effect:

k_germ(T) = k_ref × exp(-Ea/R × (1/T - 1/Tref))

Where:

• Ea = activation energy (50-70 kJ/mol)
• R = gas constant (8.314 J/mol·K)
• T = absolute temperature (K)
• Tref = reference temperature (293 K, 20°C)

Moisture Effect:

k_germ(φ) = k_max × ((φ - φmin)/(φopt - φmin))^α for φ < φopt
k_germ(φ) = k_max for φ ≥ φopt

Where:

• φmin = minimum RH for germination (typically 75-85%)
• φopt = optimal RH (typically 95-100%)
• α = moisture sensitivity exponent (2-4)

Lag Phase Duration

Pre-Germination Delay:

t_lag = t₀ × exp(A(φcrit - φ) + B(Topt - T))

Where:

• t₀ = baseline lag time at optimal conditions (1-3 days)
• A, B = empirical coefficients
• φcrit = critical RH for the substrate
• Topt = optimal temperature for growth (25-30°C)

Growth Rate Equations

Primary Model

Exponential Growth Phase:

dX/dt = μmax × X × (1 - X/Xmax) × f(T) × f(φ)

Where:

• X = biomass density (g/m²)
• μmax = maximum specific growth rate (1/day)
• Xmax = carrying capacity (g/m²)
• f(T), f(φ) = environmental response functions

Temperature Response

Cardinal Temperature Model:

f(T) = ((T - Tmin)(T - Tmax))/(((Topt - Tmin)(Topt - Tmax)) - ((Topt - T)²))

Where:

• Tmin = minimum growth temperature (0-5°C)
• Topt = optimal growth temperature (25-30°C)
• Tmax = maximum growth temperature (40-45°C)

For T < Tmin or T > Tmax: f(T) = 0

Water Activity Response

Davey Model:

f(aw) = exp(-((aw,min - aw)/k)²)

Where:

• aw = water activity (≈ RH/100)
• aw,min = minimum water activity for growth
• k = shape parameter (typically 0.1-0.2)

Secondary Metabolite Production

Mycotoxin Formation:

dTox/dt = qtox × μ × X × (1 - Tox/Toxmax)

Where:

• Tox = toxin concentration (μg/g substrate)
• qtox = specific toxin production rate (μg/g biomass·day)
• Toxmax = maximum toxin accumulation

Model Parameters

Material Properties

Biological Parameters

Computational Implementation

Time Stepping

Adaptive Step Control:

Δt_new = Δt_old × (ε_target/ε_actual)^(1/(p+1))

Where:

• ε_target = desired error tolerance (0.001-0.01)
• ε_actual = computed error estimate
• p = order of the numerical method

Spatial Discretization

Finite Volume Method:

Node spacing at material interfaces adjusted to capture moisture gradients:

• Geometric progression ratio: 1.2-1.5
• Minimum node spacing: 0.5-2 mm at surfaces
• Maximum node spacing: 10-50 mm in bulk material

Convergence Criteria

Iterations continue until:

||X^(k+1) - X^k|| < ε_abs + ε_rel × ||X^k||

Where:

• ε_abs = absolute tolerance (10⁻⁶)
• ε_rel = relative tolerance (10⁻⁴)
• X = solution vector (T, φ, M)

Validation Requirements

Models validated against:

• Laboratory growth chamber tests with controlled T and RH
• Field monitoring of building envelope assemblies
• Interstitial condensation events
• Seasonal moisture cycling in walls and roofs
• Known mold growth observations in buildings

Typical model accuracy:

• Mold index prediction: ±1 index unit
• Time to growth initiation: ±20%
• Growth rate: factor of 2-3 variation

Application Context

Biohygrothermal models serve as the foundation for:

• Building envelope design optimization
• Material selection for mold resistance
• Climate-specific construction detailing
• Retrofit analysis for moisture problems
• Long-term durability assessment
• Risk mapping for geographic regions