# 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 `Y`_{11}, W/m²K;

- external thermal admittance `Y`_{22}, W/m²K;

- periodic thermal transmittance `Y`_{12}, 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=2.718...**e** - 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
`Y`_{12}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**

Parameter | QuickField | ISO 13786:2017 | Difference, % |
---|---|---|---|

periodic penetration depth δ, m | 0.1425 | 0.144 | 1% |

thermal transmittance U, W/m²K | 3.557 | 3.56 | <1% |

internal thermal admittance Y_{11} | 5.68 W/m²K, time shift 0.93 h. | 5.70 W/m²K, time shift 0.95 h. | 2% |

external thermal admittance Y_{22} | 11.54 W/m²K, time shift 1.86 h. | 11.59 W/m²K, time shift 1.87 h. | <1% |

periodic thermal transmittance Y_{12}=Y_{21} | 1.83 W/m²K, time shift -5.65 h. | 1.83 W/m²K, time shift -5.68 h. | <1% |

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* `Y`_{12}. 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* `Y`_{21}. 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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