For bacteriological testing of water supplies and in medical...

For bacteriological testing of water supplies and in medical clinics, samples must routinely be incubated for 24 $\mathrm{h}$ at $37^{\circ} \mathrm{C}$ . A standard constant-temperature bath with electric heating and thermostatic control is not practical in war-torn places and developing countries without continuously operating electric power lines. Peace Corps volunteer and MIT engineer Amy Smith invented a low-cost, low-maintenance incubator to fill the need. It consists of a foam-insulated box containing several packets of a waxy material that melts at $37.0^{\circ} \mathrm{C}$ , interspersed among tubes, dishes, or bottles containing the test samples and growth medium (bacteria food). Outside the box, the waxy material is first melted by a stove or solar energy collector. Then the waxy material is put into the box to keep the test samples warm as it solidifies. The heat of fusion of the phase-change material is 205 $\mathrm{kJ} / \mathrm{kg}$ . Model the insulation as a panel with surface area 0.490 $\mathrm{m}^{2}$ , thickness $4.50 \mathrm{cm},$ and conductivity 0.0120 $\mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}$ . Assume the exterior temperature is $23.0^{\circ} \mathrm{C}$ for 12.0 $\mathrm{h}$ and $16.0^{\circ} \mathrm{C}$ for 12.0 $\mathrm{h}$ . (a) What mass of the waxy material is required to conduct the bacteriological test? (b) Explain why your calculation can be done without knowing the mass of the test samples or of the insulation.

For bacteriological testing of water supplies and in medical clinics, samples must routinely be incubated for 24 $\mathrm{h}$ at $37^{\circ} \mathrm{C}$ . A standard constant-temperature bath with electric heating and thermostatic control is not practical in war-torn places and developing countries without continuously operating electric power lines. Peace Corps volunteer and MIT engineer Amy Smith invented a low-cost, low-maintenance incubator to fill the need. It consists of a foam-insulated box containing several packets of a waxy material that melts at $37.0^{\circ} \mathrm{C}$ , interspersed among tubes, dishes, or bottles containing the test samples and growth
medium (bacteria food). Outside the box, the waxy material is first melted by a stove or solar energy collector. Then the waxy material is put into the box to keep the test samples warm as it solidifies. The heat of fusion of the phase-change material is 205 $\mathrm{kJ} / \mathrm{kg}$ . Model the insulation as a panel with surface area 0.490 $\mathrm{m}^{2}$ , thickness $4.50 \mathrm{cm},$
and conductivity 0.0120 $\mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}$ . Assume the exterior
temperature is $23.0^{\circ} \mathrm{C}$ for 12.0 $\mathrm{h}$ and $16.0^{\circ} \mathrm{C}$ for 12.0 $\mathrm{h}$ . (a) What mass of the waxy material is required to conduct the bacteriological test? (b) Explain why your calculation can be done without knowing the mass of the test samples
or of the insulation.

Key Concepts

Thermal Insulation

Thermal insulation is the property of a material to reduce the rate of heat transfer, maintaining a desired temperature within a confined space. Key factors include the material’s thickness, thermal conductivity, and surface area. Insulation plays a critical role in minimizing energy losses in temperature-regulated environments such as the described incubator.

Energy Balance Principles

Energy balance involves equating the energy entering a system with the energy leaving it, ensuring that all energy transfers are accounted for. In the context of the incubator, it is used to match the energy lost through insulation with the energy absorbed by the phase change material, thereby determining the required mass of that material.

Fourier's Law of Heat Conduction

This law quantifies the rate at which heat energy passes through a material due to a temperature gradient. It depends on the material's thermal conductivity, the area through which heat is transferred, and the difference in temperature across the material. In problems like this, it allows the calculation of the energy loss through the insulating panel over time.

Phase Change Materials

Phase change materials store and release thermal energy as they undergo transitions between solid and liquid states. They absorb heat when melting and release heat when solidifying, which allows them to regulate temperature effectively. Their use in designs such as incubators leverages this property to maintain a steady temperature without continuous energy input.

Latent Heat of Fusion

Latent heat of fusion is the energy required to change a unit mass of a material from solid to liquid at its melting point, without changing its temperature. This property is central in determining how much energy can be absorbed by the phase change material during melting, which in turn dictates the mass necessary to maintain the desired temperature over a specified period.

Transcript

00:01 So they were looking at the physics of a waxy material that was specially developed to keep bacteriological testing supplies warm in an insulated box.

00:10 And what they needed to do was they needed to keep these supplies at a temperature of t equals 37 .0 degrees celsius for a time of 24 hours.

00:23 And basically the idea was they created this special substance that had a melancholy.

00:30 Temperature t -s -m, also 37 .0 degrees celsius, and a latent heat of fusion of 205 kilojoules per kilogram.

00:44 And basically the idea was they would melt this waxy material and pour it into the box with the samples.

00:53 And the energy released from eventually the total mass of this waxing material fusing back into a solid was used to keep the samples warm for the 24 hour period.

01:08 So basically all they needed to do was get enough of the liquid material such that it would produce more heat than the box would just give off from radiation over the 24 hour period.

01:23 So we also want to, we're given how we want to, we're going to model the installation just as a panel with an area sub -insulation of 0 .490 square meters and a thickness sub -insulation of 0 .0 -4 -5 -0 meters, 4 .50 centimeters, and a thermal conductivity k -sub insulation of 0 .0 -120, and that's in 1 .5 .0.

01:57 Watts per meter degrees celsius.

02:02 And we're also given that the temperature outside of the box is going to be temperature sub 1 of 23 .0 degrees celsius for the first 12 hours.

02:16 And we are given that there's going to be second temperature, t sub 2, of 16 .0 degrees celsius for another 12 hours.

02:25 So again, delta t equals 12 hours.

02:29 And so basically what we want to do is we want to know that we want to keep these test samples basically isofermal with just no change in temperatures.

02:41 So we need there to be just all the heat out is equal to the heat in.

02:48 So we can go ahead and write that as the heat lost by the waxing material due to its fusion is equal to.

03:02 Basically we'll just do the radiation the power of the box during time interval 1 times delta t plus the power during the second temperature in time integral power sub 2 delta t because just power times time is energy so let's go ahead and expand these so we're going to use equation 20 .7, which gives us the energy required for a phase change, which is plus or minus depending on if it's gaining or losing energy, just the mass times the latent heat of whatever transformation is going.

03:54 And over here, we're going to use equation 20 .18, which gives us they power radiated through a slab of insulation of just a times the cross -sectional area, times the difference between the hot temperature and the cold temperature over the sum of the r values, which are just the thickness over the conductivity.

04:26 Recall r -sup i equals the sum over i of l -chi.

04:32 Of i over case of i so we can go ahead and scroll down a bit expand these terms so negative q fusion is going to be positive the mass of the waxing material times the latent heat of fusion and it's positive because it would just be a negative m out here because the liquid material is losing energy as it goes back to a solid but negative times negative is positive so here we have positive the mass that we're solving for.

05:06 We want to know how much waxing material we need, just times that latent heat of fusion from above.

05:13 And that's equal to, we're going to pull out some variables from this, from equation 20 .18, specifically, we'll grab that cross -sectional area over our r values, so thickness over the conductivity.

05:30 And that's, we'll also pull out delta t.

05:33 Because it applies to both powers...

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