## The Thermal Design – Part 2

I discussed my overall static thermal design of my house in Part I.   In this second post, I wanted to discuss the dynamic characteristics of its design – that is how the house will respond over time as external temperatures vary, but I’ve decided to break this into two parts leaving how I propose to control the internal temperature to a later post.   Again, this content is a self-developed analysis, because I have yet to find any decent design guides to build upon – the Internet either seems to be full of qualitative hand-waving at one extreme or the reader requires degree-level maths/engineering to understand the content at the other.  In my view any intelligent reader with at least O level Maths (or GCSE A-C in new money) should be able to understand the general principles here and make informed quantitative decisions relating their application in design.  So in this post, I will try to apply this to my own design.  I will accept any constructive feedback and will update this accordingly.

I look at rates of heat loss and heat gains in steady state when doing the static design – that is I assume that nothing varies of time.  Such heat losses are typically a summation of factors which relate to an intrinsic property of the house fabric known as the Thermal Resistance; this relates to the extent to which a given material resists a heat flow when a temperature difference exists across it, though conventionally its inverse, thermal conductance is usually quoted in the case of sheet materials and is usually referred to as the U value, measured in W/m²K (where K refers to degrees Kevin; the same as per °C when measuring temperature deltas).

By way of an example, let’s say that I want to maintain my house at a target temperature of 20°C.  If the external temperature is 10°C then the U value calculations that I showed in Part I give a gross heat loss of 0.65 kW for the house fabric including ventilation, with this increasing by 68 W for every extra °C in delta temperature.  However, I need to offset this loss by the intrinsic heat inputs to house which includes most of the electricity usage, roughly 0.7 kW (as I discuss below).  On top of this we have 3 warm human bodies of the occupants (0.3 kW say), and any solar gain.  However, let’s assume that it is grey overcast weather so I can ignore this last factor, and so this all gives a rough gross heat input of 1 kW into the house, or if I take off the heat losses a net heat gain of roughly 0.35 kW.

Given this heat excess, the house will slowly warm up and will only stop warming up when the heat losses and heat input are in balance, that is when the temperature has risen by (350/68)°C at roughly 25°C internal temperature.   Maybe in reality, we’d have started opening windows to cool down, but my point is that we might need such manual intervention to prevent this scenario occurring even in spring or autumn.   This might be difficult for the reader to accept, but the problem with relying on our experience is that we’ve all been brought up in poorly insulated housing stock which has a net heat deficit for at least 3 seasons a year.

If I look at the trend for typical U values (in W/m²K) for house walls over built over recent decades: the 60s – 1.8; by the 80s – 1.0; the 90s – 0.5, 2010 to 0.35.  The target figure for my house is 0.12 and it will also have MVHR, so its overall heat losses are over 15 times smaller than the house that I grew up in.  So we are used to heating systems (usually automatic central heating) to top up the heat in the house when the temperature falls below a threshold, but the heat losses are so small with this type of energy-efficient design that the intrinsic heating can frequently exceed the heat losses.

With a near PassivHaus type design, any house will be in a state of heat excess at least half the time.  If we want a controlled comfortable living environment, then knowing how we will dump excess heat is just as important as generating heat.
Understanding our heat inputs

Because the whole house design is nearer equilibrium, I need to have a good understanding of what our likely heat inputs are.  These broadly fall into three categories:

• Electricity.   I have been tracking our electricity use in our current house for over 7 years.  We have Economy 7, and our daily usage is currently 10kWhr (day rate) and 6 kWhr (night rate).  We don’t use electricity for space heating (but we do top up our DHW using E7) and all of our lighting is energy efficient.  Nonetheless, this usage still rises by perhaps 2+2 kWhr in the winter quarter.  We have the usual array of electrical equipment as well as using an overnight timer-based cold-fill dishwasher and washing machine.   We are now retired, so there isn’t a noticeable weekday / weekend usage pattern.   However, our live-in son is the main consumer of electricity with his games PC, Xbox, etc. running a lot of the time.  (The only noticeable dip in usage is when he goes on holiday!) Apart from the small amount of DHW that goes down the plug hole, this electrical energy all ultimately ends up warming the air or the fabric of the house.  So this average heat input in our case varies from roughly 650W in the summer to 850W in the winter.
• Body heat.   This varies according to body mass and activity levels but is as low as 70W when sleeping, maybe 100W when sedentary and up to double this if moderately active.  So a reasonable estimate for 3 lazy house occupants is 300W.   Small but worth including in the totals.
• Solar gain.   The energy in direct sunlight is roughly 1kW/m² directly facing the sun.   If you want to estimate the overall solar gain, then you can use the PVGIS calculator.   It’s main purpose is to predict PV array outputs, but it also gives you in one of its output columns Hd, the average daily sum of global irradiation per square meter, by calendar month.   You can plug in your location and the area (of glass) and orientation of the windows on each wall to get an average solar irradiation.   Your need to allow for any shading and the fact that windows have roughly 70% transmittance at the sun’s colour temperature.   Here is the predicted plot for my windows.

I feel that it is quite difficult for us to vary these without investment or material lifestyle changes.  Of course we will look at the energy efficiency of new appliances, PCs, etc. for the new house, but given that we’ve eliminated most of the “low hanging fruit” over the last 5 years, I think that we will have a comparable run rate in the new house.  If you are doing this same exercise, then I recommend that you do your own, albeit similar analysis, as your usage and patterns might well be very different.

The approach that I discussed for solar gains is useful to determine ballpark values to feed into your overall heating calculations.   Yes, it is intermittent and unpredictable the UK and the actuals could vary significantly from this on a daily basis.   However when I compare this graph with ones that Damon published for his actual intra-day energy collected (Earth Notes: Grid-tie PV Power) and look at the spread against his predicted (which I did by analysing his CSVs), the actual daily figures nearly always vary between 0 to 2x this predicted value with a 1σ of roughly 0.4x so this still a reasonable approximation for heat calculation purposes.

The Role of Thermal Inertia

What this analysis doesn’t tell me is how quickly the house will react to this net heat excess and how I can control this.  To do this I need to consider the impacts of a second material property usually called the thermal inertia or mass (but more strictly is the Thermal Capacity) and which relates to its ability to store heat energy.   It is normally quoted in J/KgK for bulk materials, but since I prefer to calculate everything based on lengths, areas and volumes, I find it easier just to multiple this figure by the density of the material to give a per unit volume equivalent, that is in J/m³K.   You can look these coefficients for standard materials up on the Internet (e.g. by Googling “thermal capacity for concrete”, etc. I also find the Engineering Toolbox site a useful resource for this).

These properties can very dramatically for different materials: for example EPS has a high thermal resistance but low volumetric capacity; in contrast, concrete has a poor resistance but reasonably high volumetric capacity.  Thermal resistance is linked to flow of heat, that is power or the rate of energy usage – measured in watts (W), whereas the capacity is linked to total energy stored – measured in joules (J) so any equations linking the two involved multiplying in a time factor.

In my example, if I assume that the excess heat is transferred to the internal air in the first instance and from this to the slab (as discussed in this post), the slab will exchange heat with the air at roughly 0.7 kW per °C temperature difference between the slab and the room air.  So once the overall air temperature is roughly half a degree warmer than the slab, the air and the slab will rise in temperature together, lock-stepped at this offset.   The slab is the major component in the thermal capacity of the house, and concrete has a thermal capacity of 1.9 mJ/m³K, so a 0.35 kW excess will raise the slab 1°C in 7 × 1,900,000 / 350 secs or in roughly 10 hours, so it will take days rather than hours to reach this 25°C endpoint.

The ratio of the thermal capacity to thermal conductivity for a given body, say one m² of external frame and wall has the units (J/K) / (W/K) which is in seconds when you do the cancelling out.   This is known as the thermal time constant of the body (often referred to by the Greek t, tau or τ).   This has a well defined meaning (as link explains) for homogeneous materials, but you can think of this much as the half life is used in radio-active materials: in this case its a measure of how quickly a change is temperature at one side starts to be reflected at the far side.

The τ value of my house walls is about 16 days.   What this means in practice is that I don’t need to worry about the daily variation in external temperatures or even the odd really cold or hot day when calculating expected heat losses or gains.   The fabric of the house smooths out all of this sort of variation long before it is felt internally.   (I discuss this further in my later post Modelling Thermal Lag).