A more realistic approach to the solar greenhouse of Example...

A more realistic approach to the solar greenhouse of Example 16.7 considers the time dependence of the solar input. A function that approximates the solar input is $\left(40 \mathrm{Btu} / \mathrm{h} / \mathrm{ft}^{2}\right) \sin ^{2}(\pi t / 24),$ where $t$ is the time in hours, with $t=0$ at midnight. Then the greenhouse is no longer in energy balance, but is described instead by the differential form of Equation 16.3 with $Q$ the time-varying energy input. Use computer software or a calculator with differential-equation-solving capability to find the time-dependent temperature of the greenhouse,

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives, describing how the state of a system changes over time. In many physical applications, such as modeling temperature changes, they provide a framework to represent dynamic processes and their rates of change, allowing predictions about system behavior under different conditions.

Energy Balance

The concept of energy balance involves accounting for the energy entering and leaving a system to ensure that any changes in stored energy are properly explained. In thermal systems, this principle is used to model how various inputs, like solar radiation, and losses determine the temperature or energy state over time.

Time-Dependent Forcing Functions

Time-dependent forcing functions are inputs to differential equations that change with time and drive the system's behavior. They are used to model phenomena where the driving forces, such as solar input that varies with the day, exhibit periodic or otherwise time-varying characteristics, influencing the state of the system differently at different times.

Numerical Methods for Differential Equations

Numerical methods for solving differential equations provide techniques to approximate solutions when analytic methods are either too complex or impossible. These methods, implemented through computer software or calculators, enable the simulation of real-world systems by stepping through time and calculating changes iteratively based on the governing equations.

Initial Conditions

Initial conditions specify the state of a system at the beginning of the time period under consideration. In the context of differential equations, they are crucial for determining a unique solution that accurately reflects the behavior of the system under study, ensuring that the model aligns with observed or expected starting values.

Transcript

00:01 In this problem, we are told that the average rate at which solar energy meets earth reaches earth is equal to 960 watts per meter squared.

00:17 And we're told that we can scale this to the other planets using the inverse square of the planet's distance from the sun.

00:26 So essentially the solar average rate at which solar energy reaches some other planet x will be equal to the radius of earth squared divided by the radius, not the radius of earth.

00:44 The distance between the sun and the earth squared divided by the distance between the sun and that planet squared all times s for the earth.

00:57 And so the question is using this.

01:00 Can we approximate the temperature on mars and on venus? so in order to figure this out, i actually utilized the application on global warming that was referenced in the problem.

01:19 And there we see that pi times the radius square.

01:28 Squared times s is equal to sigma times four pi, the radius squared temperature to the fourth.

01:40 And we can then go ahead and solve for temperature and you would get that temperature is equal to s over four sigma to the one fourth, where sigma is the stefan boltzman constant and is equal to 5 .67 times 10 to the negative 8 watts per meter squared kelvin to the 4th.

02:12 So for mars, we can go ahead and solve again replacing x with mars.

02:30 So for this s, we're actually going to be plugging in the radius of earth squared.

02:40 Over the radius of for mars squared.

02:44 And again, i keep saying radius because i'm using r.

02:47 It shouldn't be the radius.

02:48 It's the distance from the sun...

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