Can $R$ value be estimated by change in temperature over time?

Question

Suppose a closed sphere with a hull and an $R$-value of $10$ m² K/W and a radius of $1$ m. Suppose the temperature inside of the sphere is Ti, the temperature outside is To. The inside temperature is kept at Ti with a heat pump.

A common way to measure R-value is to assume $T_i \gg T_0$. The thin layer of air causes a slightly different temperature at the inside surface of the sphere, $T_f$. The $R$-value of such air layer is known and hence: $R_{air} \ / \ (T_i-T_f) = R \ / \ (T_f-T_0)$ and can be solved for $R$. But this requires $T_i >> T_0$.

Now, suppose in $I$ plot $T$ versus $t$ (time). When the heat pump turns off, the sphere will be slowly getting colder as heat moves out. If I wait long enough $T_i = T_0$. The speed at which this occurs should depend on the $R$ value (and surface area).

So, I am wondering, given $T(t)$ and/or $∆T /∆ t$, can I estimate the sphere's $R$-value? If yes, how? If not, why not?

Answer 1

Here's the idea, drawn in the analogy of an electrical network

setup:

• solid sphere (or spherical shell with negligible thermal mass inside it), with very high thermal conductivity compared to everything else
• surrounded by insulation with negligible thermal mass
• surrounded by air

variables:

• sphere has specific heat $c$ , and mass $m$
• conduction thru the insulation characterized by $R_1$, and effective area $A_1$
• convection between insulation and outside air characterized by $R_2$ and effective area $A_2$ (i.e. is linear vs ${\Delta}T$, perhaps questionable)
• no other significant thermal resistances

temperatures:

• $T_i$ = inside temp
• $T_o$ = outside air temp
• $T_x$ = temp at representative point (taking into account bouyancy effects etc) of boundary layer

discussion:

• If you know $T_i$ , $T_o$ , and $T_x$ , then you can compute the ratio of the R's, with a single measurement.
• If you know $T_i$ , $T_o$ , and $(c.m)$ , then you get the heat flow in the loop, and you can compute the sum of the two R's. Still with a single measurement. Good if $T_x$ were troublesome to measure.
• If you know $T_o$ , $T_x$ , and $(c.m)$ , then you can observe the exponential decay's time constant with multiple measurements, and from that compute the sum of the two R's. Good if $T_i$ cannot be measured for some reason. Also in theory it should be less sensitive to systematic error measuring $T_x$, as may be the case if we don't know exactly where to find a "representative point" on the sphere.