Transient Conduction

Transient conduction describes heat transfer when temperature varies with both position and time. This analysis is essential for HVAC applications including equipment startup, building thermal response, and thermal storage systems.

Governing Equation

The general transient heat conduction equation:

$$\frac{\partial T}{\partial t} = \alpha \nabla^2 T + \frac{\dot{q}}{k}$$

Where:

• α = thermal diffusivity (m²/s)
• ∇² = Laplacian operator
• q̇ = internal heat generation per unit volume (W/m³)

Thermal Diffusivity

Thermal diffusivity governs the rate of temperature propagation through a material:

$$\alpha = \frac{k}{\rho c_p}$$

Physical Interpretation:

• High α: rapid temperature response (metals)
• Low α: slow temperature response (insulation)

Lumped Capacitance Method

The lumped capacitance approach assumes uniform temperature throughout an object at any instant. Valid when internal thermal resistance is negligible compared to external convective resistance.

Governing Equation

Energy balance for a lumped system:

$$\rho V c_p \frac{dT}{dt} = -hA_s(T - T_\infty)$$

Solution:

$$\frac{T(t) - T_\infty}{T_i - T_\infty} = e^{-t/\tau}$$

Where the thermal time constant is:

$$\tau = \frac{\rho V c_p}{hA_s} = \frac{mc_p}{hA_s}$$

Time Constant Significance

The time constant τ represents the time required for the temperature difference to decrease to 36.8% (1/e) of its initial value.

Design Guidelines:

• τ < 5 min: rapid response (thermostats, sensors)
• τ = 15-60 min: moderate response (hydronic terminal units)
• τ = 2-8 hr: slow response (building thermal mass)
• τ > 12 hr: very slow response (earth coupling)

Biot Number

The Biot number determines the validity of the lumped capacitance assumption:

$$Bi = \frac{hL_c}{k} = \frac{\text{Internal conduction resistance}}{\text{External convection resistance}}$$

Where Lc is the characteristic length:

$$L_c = \frac{V}{A_s}$$

Criteria:

• Bi < 0.1: lumped capacitance valid (< 5% error)
• Bi < 0.05: excellent accuracy (< 2% error)
• Bi > 0.1: spatial temperature gradients significant

HVAC Applications

Fourier Number

The Fourier number is the dimensionless time parameter:

$$Fo = \frac{\alpha t}{L_c^2}$$

Physical Meaning:

• Ratio of heat conducted to heat stored
• Measures depth of thermal penetration
• Critical for transient analysis validity

Heisler Charts

Heisler charts provide graphical solutions for one-dimensional transient conduction in simple geometries when Bi > 0.1.

Available Geometries

Plane Wall (thickness 2L):

$$\frac{T(x,t) - T_\infty}{T_i - T_\infty} = f(Bi, Fo, x/L)$$

Infinite Cylinder (radius ro):

$$\frac{T(r,t) - T_\infty}{T_i - T_\infty} = f(Bi, Fo, r/r_o)$$

Sphere (radius ro):

$$\frac{T(r,t) - T_\infty}{T_i - T_\infty} = f(Bi, Fo, r/r_o)$$

Chart Application Procedure

1. Calculate Biot number: Bi = hLc/k
2. Calculate Fourier number: Fo = αt/Lc²
3. Read centerline temperature from Chart 1
4. Read position correction from Chart 2 (if needed)
5. Read total heat transfer from Chart 3 (if needed)

Limitations

• Constant properties
• Uniform initial temperature
• Constant surface conditions
• Fo > 0.2 for accuracy

Semi-Infinite Solids

A semi-infinite solid extends infinitely in one direction. Valid when thermal penetration is small compared to physical dimensions.

Penetration Depth

Approximate thermal penetration depth:

$$\delta \approx 2\sqrt{\alpha t}$$

Criterion for Semi-Infinite Assumption:

Physical dimension > 4√(αt)

Constant Surface Temperature

For sudden change to surface temperature Ts:

$$\frac{T(x,t) - T_s}{T_i - T_s} = \text{erf}\left(\frac{x}{2\sqrt{\alpha t}}\right)$$

Where erf is the error function.

Surface Heat Flux:

$$q’’_s(t) = \frac{k(T_s - T_i)}{\sqrt{\pi \alpha t}}$$

Constant Surface Heat Flux

For imposed heat flux q’’s:

$$T(x,t) - T_i = \frac{2q’’_s\sqrt{\alpha t/\pi}}{k}\exp\left(-\frac{x^2}{4\alpha t}\right) - \frac{q’’_s x}{k}\text{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right)$$

Convective Boundary Condition

For convection to fluid at T∞:

$$\frac{T(x,t) - T_i}{T_\infty - T_i} = \text{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right) - \exp\left(\frac{hx}{k} + \frac{h^2\alpha t}{k^2}\right)\text{erfc}\left(\frac{x}{2\sqrt{\alpha t}} + \frac{h\sqrt{\alpha t}}{k}\right)$$

HVAC Applications

Ground Temperature Calculations:

• Buried pipe analysis
• Earth-coupled heat exchangers
• Slab-on-grade heat loss
• Foundation heat transfer

Validity Check:

• Daily cycles: depth > 0.5 m typically valid
• Annual cycles: depth > 5 m typically valid

Building Thermal Mass

Thermal mass describes the capacity of building materials to store and release heat, dampening temperature swings.

Effective Thermal Mass

For periodic heating/cooling cycles:

$$\delta_m = \sqrt{\frac{\alpha P}{\pi}}$$

Where P is the period (24 hours for diurnal cycles).

Effective Mass Depth:

Thermal Admittance

The ability of a surface to exchange heat with the environment:

$$Y = \sqrt{k\rho c_p \omega}$$

Where ω = 2π/P is the angular frequency.

Decrement Factor

Ratio of output to input temperature amplitude:

$$f = \frac{|T_{\text{indoor amplitude}}|}{|T_{\text{outdoor amplitude}}|}$$

For massive construction: f = 0.2-0.4 For lightweight construction: f = 0.6-0.9

Time Lag

Phase shift between outdoor and indoor temperature peaks:

$$\phi = \sqrt{\frac{P}{\pi \alpha}}L$$

Design Implications:

• High thermal mass: reduced peak loads, increased time lag
• Optimal for climates with large diurnal temperature swings
• Requires night ventilation or night setback for effectiveness

Numerical Methods

For complex geometries and boundary conditions, numerical methods are required.

Finite Difference Methods

Discretize space and time domains:

$$\frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta x)^2}$$

Explicit Method:

• Simple to implement
• Stability criterion: Fo ≤ 0.5

Implicit Method:

• Unconditionally stable
• Requires matrix solution
• Crank-Nicolson provides optimal accuracy

Building Energy Simulation

Modern building simulation tools (EnergyPlus, TRNSYS) solve coupled transient equations:

• Multi-layer wall conduction
• Radiation exchange
• Convection boundaries
• Internal gains
• HVAC system interaction

Components

• Lumped Capacitance Method
• Biot Number
• Fourier Number
• Heisler Charts
• Semi Infinite Solids
• Finite Difference Methods
• Crank Nicolson Method
• Explicit Methods
• Implicit Methods
• Alternating Direction Implicit
• Finite Element Methods
• Phase Change Problems
• Moving Boundary Problems
• Stefan Problem