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# Project
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# Project: 1D Heat Transfer Simulation Report
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## **Q1: Time Derivative ($\frac{\partial T}{\partial t}$)**
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This report summarizes the solution to the 1D Heat Transfer Project, focusing on the numerical solution of the Heat Equation using the Explicit Finite Difference Method.
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- This is approximated using a **forward finite difference** formula: $$\frac{T_i^{n+1} - T_i^n}{\Delta t}$$
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## Introduction to the Heat Equation
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## **Q2: Second Spatial Derivative ($\frac{\partial^2 T}{\partial x^2}$)**
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The general 3D Heat Equation describes temperature distribution over time and space:
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- This is approximated using a **central finite difference** formula: $$\frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta x)^2}$$
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$$
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\rho c_p \frac{\partial T}{\partial t} = \lambda \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right)
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$$
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## **Q3: Reformulate the Equation**
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- Substitute the finite difference approximations into the 1D Heat Equation $ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $:
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$$\frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta x)^2}$$
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Where:
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## **Q4: Express $ T_i^{n+1} $**
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* $T$: Temperature ($^{\circ}C$)
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- Rearranging the reformulated equation to explicitly solve for temperature at the next time step:
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$T_i^{n+1} = T_i^n + \alpha \frac{\Delta t}{(\Delta x)^2} (T_{i+1}^n - 2T_i^n + T_{i-1}^n)$.
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- This can be simplified by defining $r = \alpha \frac{\Delta t}{(\Delta x)^2}$:
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$T_i^{n+1} = T_i^n + r (T_{i+1}^n - 2T_i^n + T_{i-1}^n)$.
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* $\rho$: Mass density ($kg/m^3$)
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## **Q5: Boundary and Initial Conditions**
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* $c_p$: Specific heat capacity ($J/(kg \cdot ^{\circ}C)$)
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- **Initial Condition ($T_0$):** Defines the temperature of the _entire bar (internal nodes)_ at the starting time ($t=0$). For question 7 of this project, it's $0^{\circ}C$.
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- **Boundary Conditions ($T_g$, $T_d$):** Define the fixed temperatures at the _ends of the bar_ ($x=0$ and $x=L$) for all times ($t>0$). These are $T_g=100^{\circ}C$ and $T_d=50^{\circ}C$ in question 7 of this project. These values are applied directly to the boundary nodes and are not updated by the finite difference equation.
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* $\lambda$: Thermal conductivity ($W/(m \cdot ^{\circ}C)$)
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* $x, y, z$: Spatial dimensions (m)
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* $t$: Time (s)
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For a 1D bar, assuming homogeneous and isotropic material and insulated sides (except for the ends), the equation simplifies to:
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$$
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\rho c_p \frac{\partial T}{\partial t} = \lambda \frac{\partial^2 T}{\partial x^2}
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$$
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This can be rewritten using thermal diffusivity $\alpha = \frac{\lambda}{\rho c_p}$:
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$$
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\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}
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$$
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## Project Questions and Solutions
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The following sections address the specific questions posed in the project document.
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### Question 1: Express the left term of the equation (\*) using numerical differentiation.
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The left term is the first partial derivative of temperature with respect to time, $\frac{\partial T}{\partial t}$.
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Using a forward finite difference approximation for the time derivative at node $i$ and time step $n$:
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$$
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\frac{\partial T}{\partial t} \approx \frac{T_i^{n+1} - T_i^n}{\Delta t}
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$$
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This approximates the rate of temperature change at a specific spatial point $i$ from time $t_n$ to $t_{n+1}$.
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### Question 2: Express the right term of the equation (\*) using numerical differentiation.
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The right term is the second partial derivative of temperature with respect to space, $\frac{\partial^2 T}{\partial x^2}$.
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Using a central finite difference approximation for the second spatial derivative at node $i$ and time step $n$:
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$$
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\frac{\partial^2 T}{\partial x^2} \approx \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta x)^2}
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$$
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This relates the curvature of the temperature profile at node $i$ to the temperatures of its immediate neighbors at the current time step.
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### Question 3: Reformulate the equation (\*) using the results of Question 1 and 2.
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Substitute the finite difference approximations into the 1D Heat Equation:
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$$
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\frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta x)^2}
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$$
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Where $\alpha = \frac{\lambda}{\rho c_p}$ is the thermal diffusivity.
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### Question 4: From the new formula of the equation, express $T_i^{n+1}$.
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Rearranging the reformulated equation to explicitly solve for the temperature at the next time step $T_i^{n+1}$:
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$$
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T_i^{n+1} = T_i^n + \alpha \frac{\Delta t}{(\Delta x)^2} (T_{i+1}^n - 2T_i^n + T_{i-1}^n)
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$$
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This is the core update equation for each interior node.
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For convenience, we define the Fourier number (or numerical stability parameter) $r$:
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$$
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r = \alpha \frac{\Delta t}{(\Delta x)^2}
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$$
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So the update equation becomes:
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$$
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T_i^{n+1} = T_i^n + r (T_{i+1}^n - 2T_i^n + T_{i-1}^n)
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$$
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### Question 5: What are the boundary conditions and initial temperature?
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a. **Temperature on the left** $x=0$**,** $T_g$**:**
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The temperature at the left end of the bar ($x=0$) is a fixed boundary condition: $T_g = 100^{\circ}C$.
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This means $T_1^n = 100^{\circ}C$ for all time steps $n$.
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b. **Temperature on the right** $x=L$**,** $T_d$**:**
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The temperature at the right end of the bar ($x=L$) is a fixed boundary condition: $T_d = 50^{\circ}C$.
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This means $T_{N_x}^n = 50^{\circ}C$ for all time steps $n$.
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c. **Initial temperature** $T_0$**:**
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The initial temperature of the *internal* nodes of the bar (at time $t=0$) is $T_0 = 0^{\circ}C$.
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This means $T_i^0 = 0^{\circ}C$ for all internal nodes $i$.
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### Question 6: Write the algorithm to calculate all temperatures at each point of the bar and at any time.
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The algorithm to solve the 1D Heat Equation using the Explicit Finite Difference Method involves these steps:
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1. **Define Parameters and Material Properties:** Set `L`, `dx`, `dt`, `Tg`, `Td`, `T0`, `rho`, `Cp`, `lambda`, and `t_final_s_sim`.
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2. **Calculate Thermal Diffusivity (**$\alpha$**):** Compute `α = λ / (ρ * cp)`.
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3. **Calculate Fourier Number (**$r$**) and Check Stability:** Compute `r = α * Δt / (Δx)^2`. For stability, `r` must be $\le 0.5$. A warning is issued if this condition is violated.
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4. **Discretize Spatial Domain:** Create a vector `x` representing the spatial nodes from `0` to `L` with step `dx`. Determine `Nx`, the total number of nodes.
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5. **Set Initial Condition:** Initialize a temperature array `T_current` of size `Nx`. Set all internal nodes to `T0`. Apply `Tg` and `Td` to the first and last nodes, respectively.
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6. **Set Up Time Loop:**
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* Create a `time_vector` from `0` to `t_final_s_sim` with step `dt`.
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* Initialize storage for `T_midpoint_over_time` (temperature at the bar's midpoint over all time) and `all_profiles`/`all_profile_times` (full spatial profiles at specific time points for plotting).
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* Capture the initial profile (`t=0`) if it's one of the `plot_time_points`.
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7. **Main Time-Stepping Loop:**
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* Iterate for each time step `n` from the start to `t_final_s_sim`.
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* **Create `T_next`:** Initialize `T_next` by copying `T_current`. This implicitly carries over the fixed boundary conditions.
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* **Apply Finite Difference Equation:** For each *internal* node `i` (from index 2 to `Nx-1`), calculate `T_next(i)` using the formula derived in Question 4: `T_next(i) = T_current(i) + r * (T_current(i+1) - 2*T_current(i) + T_current(i-1))`.
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* **Update `T_current`:** Set `T_current = T_next` for the next iteration.
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* **Store Data:** Record `T_current(mid_index)` into `T_midpoint_over_time`. If the current time matches a `plot_time_point` (using `ismembertol` with absolute tolerance), store the full `T_current` profile into `all_profiles`.
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### Question 7: Use MATLAB to solve the equation.
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The MATLAB code implements the algorithm described in Question 6 using the specified numerical values for parameters and material properties. The `fprintf` statements provide real-time output of derived constants (thermal diffusivity, Fourier number) and simulation progress.
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**Key Features of the MATLAB Implementation:**
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* **Parameter Definition:** All physical and numerical parameters are clearly defined at the beginning of the script.
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* **Stability Check:** The Fourier number (`r`) is calculated, and a warning is issued if it exceeds `0.5`, indicating potential numerical instability.
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* **Grid Initialization:** Spatial and temporal grids are set up, and the initial temperature profile is established with the specified boundary conditions.
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* **Iterative Solution:** A `for` loop efficiently advances the temperature solution through time, updating each internal node based on its neighbors' temperatures from the previous time step.
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* **Profile Capture:** Specific `plot_time_points` are strategically chosen to capture the temperature profiles during critical phases of heat transfer (early transient, middle, and approaching steady-state). The `unique()` function ensures these points are sorted and distinct.
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* **Robust Data Storage:** `ismembertol` with an explicit absolute tolerance (`dt/2`, `'DataScale', 1`) is used to accurately match simulation times with `plot_time_points` for storing profiles, avoiding floating-point comparison issues.
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* **Visualization:** The results are presented in a two-panel figure:
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* The left panel displays multiple **temperature profiles across the bar (Temperature vs. Position)** at the selected `plot_time_points`, illustrating the evolution of the temperature distribution.
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* The right panel shows the **temperature at the bar's midpoint as a function of time (Temperature vs. Time)**, providing a clear view of how a specific point heats up and approaches equilibrium.
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* **Conditional Legend:** A conditional check prevents the legend from being displayed if too many lines are plotted, improving readability.
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* **Figure Output:** The figure provides a comprehensive visualization of the simulation, demonstrating how the temperature profile changes over time until it reaches a steady-state (linear) distribution between the fixed boundary temperatures.
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An example of the generated figure (`Figure_1.png`) would be:
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