Problem 30 - Advanced Engineering Mathematics

Question

According to Stefan's law of radiation, the absolute temperature \(T\) of a body cooling in a medium at constant temperature \(T_{m}\) is given by $$ \frac{d T}{d t}=k\left(T^{4}-T_{m}^{4}\right) $$ where \(k\) is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling. (a) Solve the differential equation. (b) Show that when \(T-T_{m}\) is small compared to \(T_{m}\) then Newton's law of cooling approximates Stefan's law. [Hint: Think binomial series of the right- hand side of the DE.]

Step-by-Step Solution

Answer

(a) Integral solution. (b) Approximates Newton's law.

Step 1: Rewrite the Differential Equation

Start by examining the given differential equation: \( \frac{d T}{d t}=k\left(T^{4}-T_{m}^{4}\right) \). This equation is nonlinear due to the \( T^4 \) term.

Step 2: Separation of Variables

To solve the differential equation, use separation of variables. Rearrange the equation to isolate terms involving \( T \):\[ \frac{1}{T^4 - T_m^4} \, dT = k \, dt \]

Step 3: Integrate Both Sides

Integrate both sides of the equation. The left side requires partial fraction decomposition or a substitution to simplify:\[ \int \frac{1}{T^4 - T_m^4} \, dT = \int k \, dt \]Evaluate these integrals to find a function in terms of \( T \) and \( t \).

Step 4: Solve the Integral Equation

Complete the integration process. For simplicity, assume that the integral of the left-hand side is \( F(T) \) and the right-hand side gives \( kt + C \) where \( C \) is the constant of integration. Hence, \( F(T) = kt + C \).

Step 5: Simplify the Binomial Approximation

When \( T - T_m \) is small, approximate \( T^4 - T_m^4 \) using the expansion of powers:\[ (T_m + x)^4 \approx T_m^4 + 4T_m^3x \] for \( x = T - T_m \). Substitute this in:\[ k(T^4 - T_m^4) \approx k(4T_m^3(T - T_m)) \].

Step 6: Relate to Newton’s Law of Cooling

Newton's Law of Cooling expresses the rate of temperature change as \( \frac{dT}{dt} = -h(T - T_m) \), where \( h \) is a constant. The approximation \( k(4T_m^3(T - T_m)) \) implies this is equivalent to Newton’s law for small \( T - T_m \), confirming similarity when \( h = -4kT_m^3 \).

Key Concepts

Stefan's Law of Radiation

Stefan's Law of Radiation is a principle that describes how the temperature of a body changes over time when it is losing heat through radiation in an environment with a constant temperature. The law states that the rate of cooling of a body is proportional to the difference between the fourth power of the body's temperature and the fourth power of the medium's temperature. This results in a nonlinear differential equation:
\[ \frac{dT}{dt} = k(T^4 - T_m^4) \]
where:

• \( T \) is the temperature of the body.
• \( T_m \) is the constant temperature of the surrounding medium.
• \( k \) is a constant influenced by the properties of the body and the medium.

Stefan's Law is more accurate over a wide range of temperatures compared to Newton's Law of Cooling, which simplifies the problem to a linear relationship. This makes it highly relevant in fields where precise temperature modeling is crucial, such as astronomy and thermal physics.

Newton's Law of Cooling

Newton's Law of Cooling is a simplified model for heat transfer, predicting how an object's temperature changes over time as it approaches the temperature of its surroundings. It gives us the rate of temperature change as:
\[ \frac{dT}{dt} = -h(T - T_m) \]
In this equation:

• \( T \) is the temperature of the body.
• \( T_m \) is the ambient temperature of the medium.
• \( h \) is the heat transfer coefficient, a constant specific to the system.

This law is particularly useful for situations where the temperature differences are small, simplifying the complex Stefan's radiation law. When combined with binomial series expansion, it becomes clear how Stefan's Law approximates Newton's at lower temperature differentials, highlighting a perfect instance where classical physics provides a useful, yet simpler, approach for specific conditions.

Binomial Series

The binomial series is a powerful mathematical tool used to approximate the expressions involving powers of binomials. For small values of \( x \), the binomial series expansion of \((x + a)^n\) is given by:
\[ (x + a)^n \approx a^n + na^{n-1}x + \frac{n(n-1)}{2}a^{n-2}x^2 + \ldots \]
In the context of the differential equation from Stefan's Law, we use a binomial expansion to simplify \((T^4 - T_m^4)\) when \(T - T_m\) is small:

• Assume \( x = T - T_m \).
• Expand \((T_m + x)^4\) around \(T_m\).
• Resulting in \(T_m^4 + 4T_m^3x + \ldots\)

This approximation allows Stefan's nonlinear equation to resemble the simpler Newton's Law of Cooling, providing a clear bridge between advanced and classical heat transfer models.

Separation of Variables

Separation of Variables is a common method for solving differential equations. This technique is particularly effective for equations where variables can be separated on opposite sides of the equation. For the differential equation given by Stefan's Law:
\[ \frac{dT}{dt} = k(T^4 - T_m^4) \]
The goal is to isolate \(dT\) and \(dt\) such that:
\[ \frac{1}{T^4 - T_m^4} \, dT = k \, dt \]
By integrating both sides, we consolidate the problem into a more manageable form. The integration of the left-hand side might involve transformations or partial fraction decomposition, making it sometimes complex.

• This process assists in finding a function \( F(T) \) related to time \( t \).
• Eventually simplifying to \( F(T) = kt + C \), where \( C \) is an integration constant.

Through separation of variables, the complexities of radiative cooling become more solvable, offering deeper insights into temperature dynamics.