Week 4 - Response function method

Question 1

Outline the calculation steps of a time domain response function method

Answer 1

1. A given heat transfer equation in the time domain is transformed into a subsidiary equation in an imaginary space using a Laplace transform;
2. this subsidiary equation is solved while imposing a unit boundary condition – this gives a Unit Response Function (URF);
3. actual boundary conditions (BC) are resolved into equivalent triangular approximations;
4. this BC representation is multiplied by the URF to give the specific system response; and
5. responses from different BC/URF pairs are added to give the overall response.

Question 2

What is the principal difference between the time and frequency domain variants of the response function method?

Answer 2

The time domain approximates continuous phenomena such as flux and temperature as a series of triangular pulses.

The frequency domain approximates continuous phenomena as a series of sine waves of increasing frequency and reducing amplitude.

Question 3

Identify 3 principal assumptions underlying the Admittance Method and indicate how these assumptions will reduce accuracy

Answer 3

1. Boundary conditions are 24 hour period harmonic. This means that some climate time series might be poorly represented.
2. Problem parameters are time-invariant where in reality they will vary as a function of temperature and moisture content. This will lead to an under- or over-estimation of heat flow.
3. Solar gain is handled by solar gain factors that embody assumptions on shortwave radiation penetration through windows.

Question 4

Identify the three response factors as used in the Admittance Method and state the principal energy flow to which each relates.

Answer 4

Admittance: Energy entering a surface for each degree of temperature swing at the environmental point. - surface convection.

Decrement: Ratio of cyclic flux transmission to the steady state flux transmission. - transient conduction.

Surface Factor: Heat flux at the internal surface that is re-admitted to the internal environmental point when temperatures are held constant. - solar radiation.

Each possess a corresponding phase angle that determines the time difference between cause and effect.

Question 5

Describe the influence of the surface admittance factor on a room’s response to a temperature fluctuation. (Hint if it’s low)

Answer 5

Materials with low admittance values (if located at the innermost position of a wall) cannot readily absorb any fluctuation in room temperature (insulation products generally have lower admittance values). Such designs will tend to overheat.

Question 6

Write down the Admittance method formulas

Answer 6

1. Mean Solar Gain: Qs’ = Se * It * Ag

Mean Casual Gain: Qc’ = 1/24 * (qc1xt)+(qc2xt)+etc

Total Mean Heat Gain: Qt’ = Qs’ + Qc’

2. Mean internal environmental temperature:

Qt’ = (AgUg+Cv)(tei-tao) + (AfUf)(tei-teo)

Where Cv= 1/ (1/0.33NV) + (1/(4.8*∑A)

3. Swing in heat gain:

Qs* = Se’ * (Ip-It) * Ag

Qst* = f * Af * Uf * Teo
where Teo = Teo’ - teo

Qc* = Qc - Qc’
where Qc = (qx)1 + (qx)2 + (qx)3

Qa* = (Ag*Ug + Cv)(Tao)
where Tao = Tao’ - tao

Qt* = Qst* + Qa* + Qs* + Qc*

4. Swing in internal environmental temperature:

Tei* = Qt*/ (∑Ay + Cv
where Ay = table of all components

5. Peak internal temperature:

Tei = Tei* + tei

Question 7

Elaborate 6 steps involved in establishing a response function approach to energy systems simulation.

Answer 7

1. A given heat transfer equation in the time domain is transformed into a subsidiary equation in an imaginary space using a Laplace transform (PDE’s to ODE’s);
2. ODE’s transformed into algebraic equations
3. Subsidiary equations are solved by purely algebraic manipulations
4. Pre-calculates system response to individual heat inputs.
5. An inverse transformation is applied to obtain the solution in the time domain of the initial problem.
6. Sacrifices the non-linear, systemic and stochastic attributes.

Question 8

What is a unit response function (URF)?

Answer 8

The response of a linear, time invariant equation system to a unit excitation function and the time-series representation of the URF are the response factors.

URFs depend on design parameters and assumptions regarding thermo-physical properties.

The number of URFs depends on the combinations of excitation function and responses of interest.

Question 9

List the admittance method overheating assessment procedure.

Answer 9

1. determine mean heat gains from all sources;
2. calculate mean internal temperature;
3. determine mean–to–peak swing in heat gains;
4. calculate swing in internal temperature; and
5. determine peak internal temperature as (2) + (4).