ISO 13786-2017. Periodic thermal transmittance

QuickField simulation example

A homogeneous concrete wall with periodically changing temperature on its surface.

Problem Type
Plane-parallel problem of transient heat transfer.

Geometry

Given
Thermal conductivity λ = 1.8 W/K-m
Density ρ = 2400 kg/m³
Specific heat capacity C = 1000 J/kg-K
Inner surface resistance R_{s_inside} = 0.04 m²-K/W,
Outside surface resistance R_{s_outside} = 0.13 m²-K/W,
Period of thermal variations T=24 hours.

Task
Calculate following parameters and compare them with reference values from ISO 13786:2017:
- periodic penetration depth δ, m
- steady-state thermal transmittance U, W/m²K;
- internal thermal admittance Y₁₁, W/m²K;
- external thermal admittance Y₂₂, W/m²K;
- periodic thermal transmittance Y₁₂, W/m²K;

Solution
Convection coefficient α is used to define the convection boundary condition in QuickField. Convection coefficient is equal to the reciprocal of the heat transfer resistance R_s: α = 1 / R_s.
In QuickField the trigonometric functions arguments are specified in degrees, time is specified in seconds. Sinusoidal temperature can be specified by the sine formula: 1 * sin (2 * 180 * t / 86400), where t is a time variable, T=86400 is a period of thermal variations in seconds. To reach quasi-dynamic state ten periods are simulated, the measurements are made at the last period.
The ISO 13786: 2017 standard defines following conditions for calculating a particular parameter:

• The periodic penetration depth δ is a depth at which the amplitude of the temperature variations is reduced by the factor of e=2.718...
• The steady state thermal transmittance U is defined in ISO 7345 as the heat flow rate [W] in the steady state divided by area [m²] and by the temperature difference [K] between the surroundings on each side of a system:
  U = Heat flow rate / Area / Temperature difference.
• Thermal admittance Y is the amplitude of the density of heat flow rate on one side resulting from a unit temperature amplitude on the same side, when the temperature amplitude on the other size is zero.
• Periodic thermal transmittance Y₁₂ is amplitude of the density of heat flow rate on one side when the temperature amplitude on that side is zero and there is a unit temperature amplitude on the other side.

Results

In transient heat transfer problem a temperature boundary condition with sinusoidally changing temperature is applied to a surface of a semi-infinite body. Amplitude of periodic temperature variation is 1 K. At a distance of δ = 0.1425m from the surface the amplitude of a temperature fluctuation is 0.367°C = 1/e. Periodic penetration depth is 0.1425 m.

Steady-state heat transfer problem is simulated with convection boundary condition assigned to the opposite wall sides. Air temperature inside is 1°C, air temperature outside is 0°C. Calculated heat flux is 0.1778 W per 0.05 m² of the wall surface area. Steady-state thermal transmittance value is U = 0.1778W/0.05m²/1K = 3.557 W/m²K.

In transient heat transfer problem a convection boundary condition with sinusoidally changing air temperature is applied to the inside surface of the wall. Amplitude of periodic temperature variation is 1 K. At the outside surface convection boundary condition with zero temperature is applied. Ten periods were simulated to reach quasi-dynamic state. Heat flux vs. time is measured at the inside surface. The amplitude of the internal thermal admittance is 0.284W / 0.05m² = 5.68 W/m². The heat flux peak is observed at the moment of time 795840 seconds. The temperature peak occurs at the moment of time (9 + 1/4) * T = 799200 seconds. Time delay between maximum of the heat flux and maximum of the temperature is 799200-795840 = 3360 seconds (0.93 hours).

Same heat transfer problem is used to calculate periodic thermal transmittance from the inside Y₁₂. Now the heat flux vs. time is measured at the outside surface. Periodic thermal transmittance at the inside surface amplitude is 0.09157W / 0.05m² = 1.83 W/m². The heat flux peak is observed at the moment of time 819540 seconds. The temperature peak occurs at the moment of time (9 + 1/4) * T = 799200 seconds. Time delay between maximum of the heat flux and maximum of the temperature is 799200-819540 = -20340 seconds (-5.65 hours).

In transient heat transfer problem a convection boundary condition with sinusoidally changing air temperature is applied to the outside surface of the wall. Amplitude of periodic temperature variation is 1 K. At the inside surface convection boundary condition with zero temperature is applied. Ten periods were simulated to reach quasi-dynamic state. Heat flux vs. time is measured at the outside surface. The amplitude of the external thermal admittance is 0.577W / 0.05m² = 11.54 W/m². The heat flux peak is observed at the moment of time 792480 seconds. The temperature peak occurs at the moment of time (9 + 1/4) * T = 799200 seconds. Time delay between the maximum of the heat flux and maximum of the temperature is 799200-792480 = 6720 seconds (1.86 hours).

Same heat transfer problem can be used to calculate periodic thermal transmittance from the outside Y₂₁. Now the heat flux vs. time is measured at the inside surface. Periodic thermal transmittance at the outside surface amplitude is 0.09157W / 0.05m² = 1.83 W/m². The heat flux peak is observed at the moment of time 819540 seconds. The temperature peak occurs at the moment of time (9 + 1/4) * T = 799200 seconds. Time delay between the maximum of the heat flux and maximum of the temperature is 799200-819540 = -20340 seconds (-5.65 hours).

Reference:
* ISO 13786:2017 Thermal performance of building components — Dynamic thermal characteristics — Calculation methods.

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