Thermal Time Constant

[MortarThePoint]

I've been thinking about rafter insulation and diurnal temperature considerations. This has lead to to consider what the time constant of the whole house is. Calculate the sum of insulated mass heat capacity and divide it by the total heat loss rate.

Q = (SUM[ A.U.dT ] + SUM[air change heat loss]) / SUM[ m.c]

dT / dt = Q / c.m

dt = (c.m / Q) * dT

Considering a simplified example system of: (I know the proportions are silly, but its helpful to keep the numbers simple and the aspect ration about right)

• Cuboid house 20m(l) x 10m(w) x 10m(h) --> total A = 1000m2, A_walls = 600m2, A_floors = 400m2 (ignore heat capacity of ceilings)
• Average U of 0.20 W/m2K --> 4kW @ dt=20C (note thermal bridges rolled into this)
• PIV or MEV dominated air leakage of 100l/s --> 2.4kW @ dt = 20C
• Total heat loss at dT = 20C is 6.4kW
• Floors: Assumed concrete floors (inc screed) 300kg/m2 --> m_floors = 400m2 * 300kg/m2 = 120t
• Walls: Assumed blockwork inner leaf (inc plaster) 140kg/m2 --> m_walls = 130% * 600m2 * 140kg/m2 = 109t (-10% windows, +40% for internal walls)
• Concrete, c is around 1kJ/kgC = 1MJ/tC
• SUM[ m.c ] = 229t * 1MJ/tC = 229 MJ/C = 64 kWhr/C, so takes 57 kWhr to change the temperature by 1C [3.6MJ = is 1kWhr]
• The time to loss 1C at dT = 20C is 64kWhr / 6.4kW ~ 10hours.
• In exponential terms, the time constant is 200 hours since dT/dt = T/200. This yields a 'temperature half life' is 200*LN[2] = 200*0.693 = 140 hours.

 

This is going to have a large impact on how the temperature changes during the day and night as the outside temperature and the heating change. For example, if you wanted to only heat your house using economy 7 electricity and only have a 1C temperature change in the dead of winter you'd want the thermal time constant per 1C at dT=20C to be about 17hours.

 

You can approximate it quite easily from your SAP and knowledge of floor and wall materials (e.g. screed, blocks and plasterboard order). The SAP has a figure in it called average "Heat transfer coefficient, W/K" which is (39) in a STORMA 2012 Version generated SAP.

 

Time Constant in hours = 273 * (insulated mass of concrete equivalents in tons) / (heat transfer coefficient in W/K)  [(1,000,000J/MJ) / (3600s/h) = 273]

 

Our design looks to be around a time constant of 90hours, so 4.5hours to loss 1C at dT=20C or a temperature half life of 62hours.

 

This is obviously a rather simplistic approach, but I think it is useful. Would be better to include mass of timber as well.

[MortarThePoint]

To factor in timber, pine has a heat capacity of around 2500kJ/kg, much higher than concrete which surprised me.

 

Time Constant in hours =(1000* (insulated mass of concrete equivalents in tons) + 2500*(insulated mass of timber equivalents in tons))  /  3.6*(heat transfer coefficient in W/K)

 

https://www.engineeringtoolbox.com/specific-heat-solids-d_154.html

[SteamyTea]

I may come back to this when I am not so tired.

 

But I think what you have done is calculated the thermal inertia figure.

[MortarThePoint]

24 minutes ago, SteamyTea said:

I may come back to this when I am not so tired.

 

But I think what you have done is calculated the thermal inertia figure.

 

Yes effectively

 

I have also found this paper which has more detail and shows my thinking wasn't entirely crazy.

   https://www.osti.gov/etdeweb/servlets/purl/20203120

[SteamyTea]

Just read the abstract of that Swedish paper (think I quoted ones), shows how little energy a house stores in reality.

£1000 will easily by a battery and some solar panels.

 

It is basically shows what many of us already know, regardless of the material or mass of those materials, a house does not store much. These days ventilation losses tend to dominated.

[MortarThePoint]

2 minutes ago, SteamyTea said:

Just read the abstract of that Swedish paper (think I quoted ones), shows how little energy a house stores in reality.

£1000 will easily by a battery and some solar panels.

 

It is basically shows what many of us already know, regardless of the material or mass of those materials, a house does not store much. These days ventilation losses tend to dominated.

 

64kWhr per degree C is a lot. That's a Tesla per 2C. If you could effectively retrieve electricity from thermal heat, we'd have some pretty cheap solar batteries.

[SteamyTea]

3 minutes ago, MortarThePoint said:

 

64kWhr per degree C is a lot. That's a Tesla per 2C. If you could effectively retrieve electricity from thermal heat, we'd have some pretty cheap solar batteries.

Not really a lot when you consider the size of a house.  The efficiency of thermal generation plants is, at best, about 60%, but they operate at such a high temperature that it is not comparable, unless you want a house that is at 300°C.

Lithium Ion batteries are now fitted to cars, and they are coming in at about £100/kWh, not that expensive, just got to get the same mass production into home products, like the Powerwall.

There are a lot of costs when making a home system that are little to do with the battery costs i.e. a useful sized inverter, legislative compliance, packaging etc.

[MortarThePoint]

3 minutes ago, SteamyTea said:

unless you want a house that is at 300°C

 

I'd accept 300K on a warm day

 

3 minutes ago, SteamyTea said:

There are a lot of costs when making a home system that are little to do with the battery costs i.e. a useful sized inverter, legislative compliance, packaging etc.

 

I'm seriously considering having a 48V lighting system in the house because I think I should be able to design and install it myself all outside of part P and have better efficiency than OTS led lights, as well as use a 48V DC battery charged off Economy7 or solar.

[SteamyTea]

1 minute ago, MortarThePoint said:

I'd accept 300K on a warm day

 

Yes, whoops, mean °C

2 minutes ago, MortarThePoint said:

I'm seriously considering having a 48V lighting system in the house because I think I should be able to design and install it myself all outside of part P and have better efficiency than OTS led lights, as well as use a 48V DC battery charged off Economy7 or solar.

That is not a bad idea, especially now we have low power LED lights.

I would design it as a 230V system though and just run it off a cheap inverter, then it can use standard switch gear, cabling and installation methods, and easily be wired into a consumer unit if needed.

[MortarThePoint]

2 minutes ago, SteamyTea said:

I would design it as a 230V system though and just run it off a cheap inverter, then it can use standard switch gear, cabling and installation methods, and easily be wired into a consumer unit if needed.

 

I've got some grand plans in this area, but would use mains voltage compliant wiring etc everywhere so that it could be swapped back to mains if needed (e.g. future house buyer).

[TonyT]

Why make it difficult. Stick to 230 vac leds

 

run lighting cabling back to a central joint box if you want to automate, if not wire it as a joint box method.

little bit more cable used but you can get the best of both worlds should you wish to automate at a later date

 

[MortarThePoint]

4 hours ago, MortarThePoint said:

Our design looks to be around a time constant of 90hours, so 4.5hours to loss 1C at dT=20C or a temperature half life of 62hours.

 

Adding in plasterboard and timber takes that time constant to 113hours, so 5.7hours to loss 1C at dT=20C or a temperature half life of 79hours.

[MortarThePoint]

Another interesting part of this is that you can approximate the way the house will respond to cyclical changes in external temperature like the diurnal temperature change. To make an analogy with electronics, the insulation acts like a resistor and the thermal mass as a capacitor. That makes for an RC low pass filter and the RC value is the Time Constant calculated above.

The transfer function of an RC filter depends on the frequency of the input oscillation and the magnitude is:

where RC is the time constant and w = 2*PI*f is the angular frequency and f is the frequency. 

 

For a diurnal temperature change it passes through 2*PI of angle in 24hours so w = 2*PI / 24 = 0.26 [NB: period T is 24hours which is inverse of frequency]

V_in can be the temperature oscillation outside the house and V_out can be taken as the temperature oscillation inside the house [bit confusing there with in and out swapped, but that's due to the transfer function mapping an input, the changing temperature outside, to an output, the changing temperature inside].

 

If we used the time constant calculated above RC = 200hr and take the diurnal temperature change as 20C, the internal temperature change would be:

20C / SQRT[(0.26*200)^2 + 1] ~ 20C / (0.26*200) = 0.38C

For an wRC >> 1 the SQRT[(wRC)^2 + 1] = wRC. Any time constant above 16hours should be OK for that approximation.

 

STOP This neglects an important factor though, solar gains! If your house is completely shaded then look no further.

• The surface of brickwork will be hotter than the outside air temperature. Let's guess it reaches 40C and goes down to 10C over night so double the temperature.
• Windows let in the suns warming rays and so the heat input isn't just via conduction and air changes. Not immediately clear how to factor that in. This highlights the benefits of masonry construction and shutters in hot countries.

I can conceive of 5kW of peak solar gain through windows during a hot summer and this would be 5kW/3.2kW = 156% of the heat transfer due to conductivity and air changes at dT = 10C. This could be approximated by using a compensated time constant:

Q = (2*SUM[ A.U.dT ] + SUM[air change heat loss] + [Solar Gain])

SGTC = (1000* (insulated mass of concrete equivalents in tons) + 2500*(insulated mass of timber equivalents in tons))  /  3.6*(500 + heat transfer coefficient in W/K)

 

In the simplified example in the first post, Q becomes around 5.2kW + 5kW = 10.2kW at dT = 10C [NB: 6.4kW was at dT=20C and we have doubled the heat flow due to conduction]. That makes the compensated time constant:

SGTC = (64kWhr/C) / (10.2kW / 10C) = 63hours

That makes the internal temperature change 20C / (0.26*63) = 1.2C. That's the minimum to maximum.

 

[the deficiencies of all this a multiple, but it's likely to be indicative. Yes, it ignores the effect of the subfloor earth etc]

 

If our time constant was also reduced to around a third of the uncompensated value, that would make for an SGTC=40hours temperature oscillation of 1.9C min to max. I would be pretty pleased with that. The outside temperature changing from 10C to 30C and in inside temperature changing from 19C to 21C without any heat pumps involved.

 

There is an uncomfortable level of approximation in what I've just written, so if someone else knows of a simple was of doing a similar calculation that would be great.

[MortarThePoint]

8 hours ago, MortarThePoint said:

Q = (SUM[ A.U.dT ] + SUM[air change heat loss]) / SUM[ m.c]

 

That should either have been:

Q = (SUM[ A.U.dT ] + SUM[air change heat loss])

Or:

dT/dt = (SUM[ A.U.dT ] + SUM[air change heat loss]) / SUM[ m.c]

[SteamyTea]

1 hour ago, MortarThePoint said:

so if someone else knows of a simple was of doing a similar calculation that would be great.

Here is the proper way to do it, not saying it is easier.

https://en.wikipedia.org/wiki/Heat_equation

 

Right at the start, when you right dT/dt are you really meaning a differential equation?  If so, it should be δT/δt, and you would have to solve the first order differential equation and then substitute the number back in.

 

Or divide a small change by another small change, which should be ΔT/Δt and this is an ordinary equation.

[Olf]

13 hours ago, MortarThePoint said:

To make an analogy with electronics, the insulation acts like a resistor and the thermal mass as a capacitor.

As coming from that background I've always though like that, thank you for bringing it out!

 

13 hours ago, MortarThePoint said:

That makes for an RC low pass filter and the RC value is the Time Constant

Precisely! Even the symbols (R -thermal resistance, C-thermal capacity) are identical and so the result of units calculation: [K / W] x [J / K] = [W x s] / [W] =

Thinking in static (= DC) terms is easier and used for example here https://en.wikipedia.org/wiki/Thermal_resistance, but indeed it is the dynamic (=AC) that is applicable to analyse diurnal cycles.

 

Be careful, as the math works only when you substitute voltage V with heat Q, not directly with temperatures T.

 

The concept also allows to add convective losses (air infiltration) as a paraller (shunt) resistance for full analysis. With all the consequences, eg that the optimum airtightness is achieved where those losses are equal to the conductive losses through the fabrics (any further improvement to either will provide diminishing returns), or that with poor airtightness, investment in low U materials is pointless (alternatively, sealing joints of thin walls also not being very productive)

[MortarThePoint]

50 minutes ago, Olf said:

As coming from that background I've always though like that, thank you for bringing it out!

 

Precisely! Even the symbols (R -thermal resistance, C-thermal capacity) are identical and so the result of units calculation: [K / W] x [J / K] = [W x s] / [W] =

Thinking in static (= DC) terms is easier and used for example here https://en.wikipedia.org/wiki/Thermal_resistance, but indeed it is the dynamic (=AC) that is applicable to analyse diurnal cycles.

 

Be careful, as the math works only when you substitute voltage V with heat Q, not directly with temperatures T.

 

The concept also allows to add convective losses (air infiltration) as a paraller (shunt) resistance for full analysis. With all the consequences, eg that the optimum airtightness is achieved where those losses are equal to the conductive losses through the fabrics (any further improvement to either will provide diminishing returns), or that with poor airtightness, investment in low U materials is pointless (alternatively, sealing joints of thin walls also not being very productive)

 

Temperature (T) should be analogous to Voltage (V) as it drives a flow of heat (Q) which would be analogous to current (I). V = I*R, T = Q*R. Would be nice to have passive equivalents to an inductor and a diode. 

[SteamyTea]

56 minutes ago, Olf said:

coming from that background I've always though like that, thank you for bringing it out

I think if you expand all the derived units you send up with the golden three if kg, m and s, in the same places.

Newton's Law of Cooling is the same as discharging a capacitor.

Population growth formula are the same as charging up a capacitor or battery.

 

e = 1+1/1!+1/2!+1/3!+1/4!....

A harmonic series.

Edited June 6, 2021 by SteamyTea

[SteamyTea]

3 minutes ago, MortarThePoint said:

Would be nice to have passive equivalents to an inductor and a diode. 

A heat pipe is like a diode.

Not sure what an inductive would be. Shall ponder that as I wander about. Basically something that can oscillate with minimum losses I think.

[MortarThePoint]

I read of a Peltier based thermal inductor, but that's a bit of a cheat probably.

Edited June 6, 2021 by MortarThePoint

[Olf]

22 hours ago, MortarThePoint said:

Temperature (T) should be analogous to Voltage (V) as it drives a flow of heat (Q) which would be analogous to current (I). V = I*R, T = Q*R.

Agree, apologies for messing up. I must have felt subconciously that something is wrong, as somehow enabled strikethrough

 

22 hours ago, MortarThePoint said:

Would be nice to have passive equivalents to an inductor and a diode. 

I don't think 'passive thermal inductor' is possible, as that would enable an oscillator, and step further is a generator. 

Phase change produces kind of Zener diode effect.

Possibly for radiant heat only, as it is pure electromagnetism, the solution would be easier to find - but that's way out of scope of the forum

[MortarThePoint]

20 minutes ago, Olf said:

Agree, apologies for messing up. I must have felt subconciously that something is wrong, as somehow enabled strikethrough

 

I don't think 'passive thermal inductor' is possible, as that would enable an oscillator, and step further is a generator. 

Phase change produces kind of Zener diode effect.

Possibly for radiant heat only, as it is pure electromagnetism, the solution would be easier to find - but that's way out of scope of the forum

 

An oscillator can be lossy so no need to violate Physics, but I am as sceptical as you.

 

I think phase change materials (PCM) will be like a battery holding a constant voltage as the current varies.

 

A thermal diode is a thing, but currently only at very low temperature:https://en.wikipedia.org/wiki/Thermal_diode

Edited June 7, 2021 by MortarThePoint

[MortarThePoint]

23 hours ago, SteamyTea said:

A heat pipe is like a diode.

 

Yes these can act as a diode as well.