On Thanksgiving Day in St. Louis a sudden storm reduces the...

On Thanksgiving Day in St. Louis a sudden storm reduces the ambient air to ? $-15^{\circ} \mathrm{C}$. If the ground was at a uniform temperature of $20^{\circ} \mathrm{C}$ before the storm, what is the surface temperature after 4 hours, and to what depth below the surface will the freezing temperature have penetrated? Take $k=1.2 \mathrm{~W} / \mathrm{m} \mathrm{K}$, $\alpha=0.40 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, h_{c}=20 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. Ignore any phase change effects.

On Thanksgiving Day in St. Louis a sudden storm reduces the ambient air to
? $-15^{\circ} \mathrm{C}$. If the ground was at a uniform temperature of $20^{\circ} \mathrm{C}$ before the storm, what is the surface temperature after 4 hours, and to what depth below the surface will the freezing temperature have penetrated? Take $k=1.2 \mathrm{~W} / \mathrm{m} \mathrm{K}$, $\alpha=0.40 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, h_{c}=20 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. Ignore any phase change effects.

Key Concepts

Convective Boundary Conditions

Convective boundary conditions model the interaction between a solid surface and a fluid (often air) in motion. In these conditions, the heat transfer from the solid surface to the fluid (or vice versa) is characterized by a heat transfer coefficient, which relates the convective heat flux to the temperature difference between the surface and the ambient fluid. This concept is crucial when the ambient conditions change, such as a sudden cold storm affecting the surface temperature.

Thermal Diffusivity

Thermal diffusivity is a material property that measures the rate at which heat conducts through a material relative to its ability to store thermal energy. It combines the effects of thermal conductivity, density, and specific heat capacity. This concept is central in transient heat conduction problems because it governs the speed at which temperature changes propagate within the material.

Thermal Penetration Depth

Thermal penetration depth defines the distance into a material that a change in surface temperature has effectively influenced over a given period. It is a useful concept in transient conduction analysis to estimate how far the altered surface conditions (such as a cold ambient temperature) extend into the material, and it typically depends on the thermal diffusivity and the time over which conduction occurs.

Transient Heat Conduction

Transient heat conduction refers to the study of how temperature within a material changes with time when subjected to a sudden change in boundary conditions. This is an important concept in understanding the time?dependent thermal response of a medium, particularly when the temperature at the surface is suddenly altered, leading to temperature gradients that evolve over time throughout the material.

Transcript

00:01 We have this function, t of h is equal to 0 .8h squared minus 16h plus 60.

00:05 This is describing the temperature in the number of h hours after a storm began in some town, which is experiencing a major winter storm.

00:14 As you can see here, i've also graphed the function down here on the graph, as the book recommends.

00:19 And using this graph, we're going to be finding some different points on the graph, some different aspects of this function.

00:25 So first off, we need to find the temperature when the storm first began.

00:29 So when the storm began, zero hours had passed.

00:32 That means that h is equal to 0.

00:34 So what we're looking for is t of 0.

00:37 This is equal to point h, 0, 0, 0, minus 16 times 0, plus 60.

00:46 All of these zeros are just going to be gone because we're multiplying by 0, so we're just left with 60.

00:51 Thus, the temperature at the start of the storm was 60 degrees fahrenheit.

00:56 Now that was part a.

00:57 It's time for part b.

00:58 We need to figure out at what time did the temperature drop below zero.

01:03 Well, to do this, we're going to look at the graph.

01:05 You could also do this just from analyzing the equation and finding its zeros.

01:09 However, we also have the graph, which makes things much easier.

01:12 So as you can see, right here, the graph crosses the axis.

01:17 This looks like it's about at five.

01:19 Thus, we can assume that at five hours, the temperature dropped below zero.

01:24 However, we should double check this because we're just eyeballing this on a graph.

01:27 So we're going to be factoring out this equation.

01:31 Let's set it equal to zero.

01:33 If we want to figure out where a temperature dropped below zero, we just set temperature equal to zero.

01:37 So we have 0 equals 0 .8h squared minus 16h plus 60.

01:44 Now i don't like this point 8 here.

01:47 It makes things much more difficult.

01:49 Nobody likes working with decimals...

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