---
# System prepended metadata

title: Heat Transfer
tags: [Math360]

---

# Heat Transfer

This section is dedicated to modeling thermal systems. We begin with the **law of conservation of energy** to derive the **energy balance equation**, which governs how the temperature of an object changes over time due to absorbed and emitted heat. We will apply **constitutive laws**, such as the **Stefan–Boltzmann law**, to describe the components of heat transfer. Using *nondimensionalization*, we'll simplify these models to identify key relationships between parameters, preparing them for numerical solution using SciPy.

### Big Ideas
* The **law of conservation of energy** is the fundamental principle used to derive differential equations that model heat transfer.
* **Constitutive laws** (like the **Stefan–Boltzmann law** for radiation) are specialized equations used to define individual components of a physical model.
* **Nondimensionalization** is essential for simplifying models by reducing the number of parameters and determining how these parameters influence system dynamics.
* `SciPy` is utilized to compute **numerical solutions**, enabling easy simulation and visualization of thermal models.



## Energy Balance


### Heat Capacity
**Heat** is the transfer (or flux) of **thermal energy**. The **heat capacity** $C$ of an object relates the change in thermal energy $\Delta E$ to the resulting change in temperature $\Delta T$:

$$
C = \frac{\Delta E}{\Delta T}
$$ 

$C$ has units of $J/K$. Although heat capacity generally depends on temperature and pressure, we will assume $C$ is *constant* for the objects in our models.

To model how temperature changes over time, we use the derivative form:

$$
C \frac{dT}{dt} = \frac{dE}{dt}
$$

This relates the rate of change of energy (heat transfer) to the rate of change of temperature.


### Energy Balance
Applying the law of conservation of energy to an object that absorbs and emits heat yields the *energy balance equation*:

$$
C \frac{dT}{dt} = Q_{in} - Q_{out}
$$

where:
* $T$ is the temperature of the object $(K)$
* $C$ is the heat capacity ($J/K$)
* $Q_{in}$ is the rate of heat absorbed by the object ($W = J/s$)
* $Q_{out}$ is the rate of heat emitted by the object ($W = J/s$)



## Dimensions and Units
| Quantity | Symbol | Dimensions | SI Units |
| :---: | :---: | :---: | :---: |
| thermal energy | $E$ | M L<sup>2</sup> T<sup>-2</sup> | J |
| temperature | $T$ | $\Theta$ | K |
| heat capacity | $C$ | M L<sup>2</sup> T<sup>-2</sup> $\Theta^{-1}$ | J/K |
| heat | $Q$ | M L<sup>2</sup> T<sup>-3</sup> | W |



## Thermal Radiation


### Solar Radiation
**Thermal radiation** is heat emitted as electromagnetic waves. The *solar constant* $S_0$ is the thermal radiation emitted by the Sun, measured per unit area per unit time just outside the Earth's atmosphere:

$$
S_0 = 1368 \ \text{W/m}^2
$$

Viewed from the Sun, the Earth presents a cross-sectional area of $\pi R^2$ (where $R$ is the radius of the Earth). Therefore, the total energy flux arriving from the Sun at the outer edge of the atmosphere is $\pi R^2 S_0$.


### Albedo
The **albedo** $\alpha$ is a *dimensionless parameter* that measures the fraction of the Sun's radiation that is reflected by the Earth back into space ($0<\alpha<1$). The total energy flux from the Sun *absorbed* by the Earth is:

$$
(1 - \alpha) \pi R^2 S_0
$$


### Stefan–Boltzmann Law
A *black body* is an idealized object that emits and absorbs thermal radiation perfectly at all frequencies. The **Stefan–Boltzmann law** states that the total energy $Q$ radiated by a black body per unit surface area per unit time is directly proportional to $T^4$:

$$
Q = \sigma T^4
$$

where $\sigma = 5.670374419 \times 10^{-8}$ W m<sup>-2</sup> K<sup>-4</sup> is the **Stefan-Boltzmann constant**.

Assuming the Earth is a black body with homogeneous temperature $T$, the total thermal radiation emitted from its surface area $4 \pi R^2$ would be $4 \pi R^2 \sigma T^4$.


### Greenhouse Effect
The chemical composition of the Earth's atmosphere affects how much longwave radiation is emitted into space. The [**greenhouse effect**](https://en.wikipedia.org/wiki/Greenhouse_effect) occurs because atmospheric gases absorb longwave radiation from the Earth but allow shortwave solar radiation to pass through.

To model this, we introduce the **greenhouse parameter** $\varepsilon$, which is the fraction of radiation emitted by the Earth that escapes into outer space (where $1-\varepsilon$ is the fraction absorbed by the atmosphere).

The total thermal radiation emitted by the Earth *into outer space* is:

$$
4 \pi R^2 \sigma \varepsilon T^4
$$



## Global Energy Balance
We construct a mathematical model for the Earth's temperature $T$ over time using the energy balance equation.

```python
import numpy as np
import matplotlib.pyplot as plt
import scipy.integrate as spi
```


### Variables and Parameters
| Description | Symbol | Dimensions | Type |
| :---: | :---: | :---: | :---: |
| temperature of the Earth and atmosphere | $T$ | $\Theta$ | dependent variable |
| time | $t$ | T | independent variable |
| solar constant | $S_0$ | M T<sup>-3</sup> | parameter |
| albedo of the Earth | $\alpha$ | 1 | parameter |
| radius of the Earth | $R$ | L | parameter |
| Stefan-Boltzmann constant | $\sigma$ | M T<sup>-3</sup> $\Theta^{-4}$ | parameter |
| heat capacity of the Earth and atmosphere | $C$ | M L<sup>2</sup> T<sup>-2</sup> $\Theta^{-1}$ | parameter |
| greenhouse parameter | $\varepsilon$ | 1 | parameter |


### Assumptions and Constraints
* The Earth and the atmosphere are treated as one object with homogeneous temperature $T$ and heat capacity $C$.
* $S_0$, $\alpha$, $C$, and $\varepsilon$ are *constant*.
* Earth emits radiation as a black body $\sigma T^4$.


### Construction
The governing equation is the energy balance equation, $C \frac{dT}{dt} = Q_{in} - Q_{out}$:

$$
C \frac{dT}{dt} = \underbrace{(1 - \alpha) \pi R^2 S_0}_{Q_{in} \ (\text{Solor Absorption})} - \underbrace{4 \pi R^2 \sigma \varepsilon T^4}_{Q_{out} \ (\text{Space Emission})}
$$


### Nondimensionalization
We apply the procedure to simplify the equation and identify key parameters.

#### Step 1 & 2 (Variables and Substitution)
Let $t = [t] t^*$ and $T = [T] T^*$.

$$
C \frac{[T]}{[t]} \frac{dT^*}{dt^*} = (1 - \alpha) \pi R^2 S_0 - 4 \pi R^2 \sigma \varepsilon [T]^4 T^{*4}
$$

#### Step 3 & 4 (Normalize and Simplify Coefficients)
Divide by the coefficient of the $T^{*4}$ term, $4 \pi R^2 \sigma \varepsilon [T]^4$: 

$$
\frac{C}{4 \pi R^2 \sigma \varepsilon [T]^3[t]} \frac{dT^*}{dt^*} = \frac{(1 - \alpha)S_0}{4 \sigma \varepsilon  [T]^4} - T^{*4}
$$

We choose the temperature scale $[T]$ to make the constant term equal to 1, and the time scale $[t]$ to make the derivative coefficient equal to 1:

\begin{align}
[T] &= \left( \frac{(1 - \alpha) S_0}{4 \sigma \varepsilon} \right)^{1/4} \\
[t] &= \frac{C}{4 \pi R^2 \sigma \varepsilon [T]^3} = \frac{C}{4 \pi R^2 \sigma \varepsilon} \left( \frac{4 \sigma \varepsilon}{(1 - \alpha) S_0} \right)^{3/4}
\end{align}

#### Step 5 (Rewrite System and Analysis)
The final dimensionless system is independent of the parameters $C$ and $R$:

$$
\frac{dT^*}{dt^*} = 1 - T^{*4}, \quad T^*(0) = \frac{T_0}{[T]}
$$


### Analysis
Since $T>0$, there is only one steady-state solution found by setting $dT^∗/dt^∗=0$:

$$
T^*_{\infty} = 1
$$

In terms of physical quantities, the equilibrium temperature is:

$$
T_{\infty} = \left( \frac{(1 - \alpha) S_0}{4 \sigma \varepsilon} \right)^{1/4}
$$

The steady-state temperature depends only on the solar parameters $(S_0, \alpha)$ and the radiation parameters $(\sigma, \varepsilon)$. The parameters $C$ and $R$ only affect the time scale $[t]$.

```python=
f = lambda T,t: 1 - T**4
t = np.linspace(0,1,100)

for T0 in [0.5,0.75,1.0,1.25,1.5]:
    T = spi.odeint(f,T0,t)
    plt.plot(t,T,'b')

plt.ylim([0,2]),plt.grid(True)
plt.show()
```

![8f4cd003-e5e4-4ff6-87d4-ceecbae58d02](https://hackmd.io/_uploads/ry7_Nl22ex.png)