Math Problem Statement
Heat transfer from a body to its surroundings by radiation, based on the Stefan-Boltzmann law, is described by the differential equation
dudt=−α(u4−T4),
where u(t) is the absolute temperature of the body at time t, T is the absolute temperature of the surroundings, and α is a constant depending on the physical parameters of the body. However, if u is much larger than T, the solutions of the above equation are well-approximated by solutions of the simpler equation
dudt=−αu4.
Suppose that a body with initial temperature 2000 K is surrounded by a medium with temperature 180 K and that α=2.2×10−12K−3/s.Find the temperature of the body at any time by solving the simpler differential equation.
Solution
We are tasked with solving the simpler differential equation given by:
where , and .
Step 1: Separate Variables
We first separate the variables to make the equation easier to solve. Rearrange the equation as:
Step 2: Integrate Both Sides
Now, integrate both sides. On the left-hand side, we have:
The integral of is:
So, we can rewrite the equation as:
where is the constant of integration.
Step 3: Solve for
Multiply both sides by and rearrange:
To solve for , we first solve for :
Finally, take the cube root:
Step 4: Use the Initial Condition
We are given that . Substituting this into the equation for when , we get:
Solve for :
Step 5: Final Expression for
Substitute this value of into the expression for :
Substituting :
This is the temperature of the body at any time .
Would you like more details or have any questions?
5 related questions:
- How does the Stefan-Boltzmann law relate to blackbody radiation?
- How does the temperature of the surroundings influence the heat transfer process?
- What would happen if the body temperature is close to the surroundings temperature in this problem?
- How can this type of differential equation be solved numerically?
- How do the initial conditions affect the solution to differential equations like this one?
Tip:
In problems involving heat transfer, simplifying assumptions (like ) can make equations more manageable but should be carefully checked against real conditions for accuracy.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Separation of Variables
Initial Value Problems
Formulas
Stefan-Boltzmann Law: Q = σA(T^4 - T_s^4)
Simplified Differential Equation: dudt = −αu^4
General Solution for Separated Variables: ∫du/u^4 = -∫α dt
Theorems
Stefan-Boltzmann Law
Techniques of Solving Separable Differential Equations
Suitable Grade Level
Undergraduate Physics/Mathematics
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