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:

dudt=αu4,\frac{du}{dt} = -\alpha u^4,

where α=2.2×1012K3/s\alpha = 2.2 \times 10^{-12} \, \text{K}^{-3}/\text{s}, and u(0)=2000Ku(0) = 2000 \, \text{K}.

Step 1: Separate Variables

We first separate the variables to make the equation easier to solve. Rearrange the equation as:

duu4=αdt.\frac{du}{u^4} = -\alpha \, dt.

Step 2: Integrate Both Sides

Now, integrate both sides. On the left-hand side, we have:

duu4=αdt.\int \frac{du}{u^4} = \int -\alpha \, dt.

The integral of 1u4\frac{1}{u^4} is:

1u4du=13u3.\int \frac{1}{u^4} du = -\frac{1}{3u^3}.

So, we can rewrite the equation as:

13u3=αt+C,-\frac{1}{3u^3} = -\alpha t + C,

where CC is the constant of integration.

Step 3: Solve for u(t)u(t)

Multiply both sides by 1-1 and rearrange:

13u3=αt+C.\frac{1}{3u^3} = \alpha t + C.

To solve for uu, we first solve for u3u^3:

u3=13(αt+C).u^3 = \frac{1}{3(\alpha t + C)}.

Finally, take the cube root:

u(t)=(13(αt+C))1/3.u(t) = \left( \frac{1}{3(\alpha t + C)} \right)^{1/3}.

Step 4: Use the Initial Condition

We are given that u(0)=2000Ku(0) = 2000 \, \text{K}. Substituting this into the equation for u(t)u(t) when t=0t = 0, we get:

20003=13C.2000^3 = \frac{1}{3C}.

Solve for CC:

C=13×20003=13×8×109=12.4×1010.C = \frac{1}{3 \times 2000^3} = \frac{1}{3 \times 8 \times 10^9} = \frac{1}{2.4 \times 10^{10}}.

Step 5: Final Expression for u(t)u(t)

Substitute this value of CC into the expression for u(t)u(t):

u(t)=(13(αt+12.4×1010))1/3.u(t) = \left( \frac{1}{3 \left( \alpha t + \frac{1}{2.4 \times 10^{10}} \right)} \right)^{1/3}.

Substituting α=2.2×1012K3/s\alpha = 2.2 \times 10^{-12} \, \text{K}^{-3}/\text{s}:

u(t)=(13(2.2×1012t+12.4×1010))1/3.u(t) = \left( \frac{1}{3 \left( 2.2 \times 10^{-12} t + \frac{1}{2.4 \times 10^{10}} \right)} \right)^{1/3}.

This is the temperature u(t)u(t) of the body at any time tt.

Would you like more details or have any questions?

5 related questions:

  1. How does the Stefan-Boltzmann law relate to blackbody radiation?
  2. How does the temperature of the surroundings influence the heat transfer process?
  3. What would happen if the body temperature is close to the surroundings temperature in this problem?
  4. How can this type of differential equation be solved numerically?
  5. How do the initial conditions affect the solution to differential equations like this one?

Tip:

In problems involving heat transfer, simplifying assumptions (like uTu \gg T) 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