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title: Derivation of one-dimensional steady state conduction
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# Derivation of one-dimensional steady state conduction
Unit 1, Lecture 3

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## Derivation of one-dimensional steady state conduction

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 3: General Heat Conduction Equation

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Welcome. Today is a mathematical lecture. We will derive the most general form of the heat conduction equation from the first law of thermodynamics, and then see how it simplifies for practical problems.
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## Lecture Agenda

- 1. Need for a General Equation
- 2. The Control Volume and Energy Balance
- 3. Heat Flux Components (x, y, z directions)
- 4. Internal Heat Generation and Storage
- 5. General 3D Conduction Equation
- 6. Simplified Forms (Poisson, Laplace, Fourier)
- 7. Reduction to 1D Steady State Equation

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We will start by establishing an energy balance on a tiny differential element, substitute Fourier's law into it, and arrive at the general differential equation for conduction.
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## Why a General Equation?

- Fourier's Law $Q = -kA(dT/dx)$ is sufficient for simple 1D steady-state problems.
- However, real problems can involve:
- - Multi-dimensional heat flow (x, y, and z).
- - Unsteady (transient) conditions where temperature changes with time.
- - Internal heat generation (e.g., electrical heaters, chemical reactions, nuclear fuel rods).
- We need a differential equation that governs temperature distribution $T(x,y,z,t)$ in a medium.

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If we want to find the temperature at any point in a solid at any given time, we must solve a differential equation subject to boundary and initial conditions.
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## The Control Volume

- Consider a small differential element (Control Volume) in a Cartesian coordinate system.
- Dimensions: $dx$, $dy$, $dz$.
- Volume of element: $dV = dx \cdot dy \cdot dz$.
- Heat is conducted into and out of this element across its six faces.

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Imagine a tiny rectangular block inside our material. We will track all energy entering, leaving, being generated inside, and being stored inside this block.
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## Energy Balance Equation

- Based on the First Law of Thermodynamics (Conservation of Energy):
- (Rate of heat conducted IN)
- + (Rate of heat GENERATED internally)
- = (Rate of heat conducted OUT)
- + (Rate of energy STORED)
- $$Q_{in} + Q_{gen} = Q_{out} + Q_{stored}$$

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This is the fundamental principle. Energy cannot be created or destroyed. Whatever goes in, plus whatever is made inside, must either come out or stay inside, raising the temperature.
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## Heat Flow in X-Direction

- Heat entering left face (at $x$): $Q_x = -k (dy\cdot dz) \frac{\partial T}{\partial x}$
- Heat leaving right face (at $x+dx$): Using Taylor Series expansion (neglecting higher order terms):
- $$Q_{x+dx} = Q_x + \frac{\partial Q_x}{\partial x}dx$$
- Net heat accumulated in x-direction:
- $$Q_x - Q_{x+dx} = -\frac{\partial Q_x}{\partial x}dx = \frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) dx\cdot dy\cdot dz$$

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We apply Fourier's law at the face x, and then use a Taylor series approximation to find the heat leaving at face x+dx. The difference is the net heat trapped due to x-direction flow.
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## Heat Flow in Y and Z Directions

- Similarly, net heat accumulated in y and z directions:
- Y-direction:
- $$Q_y - Q_{y+dy} = \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) dx\cdot dy\cdot dz$$
- Z-direction:
- $$Q_z - Q_{z+dz} = \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) dx\cdot dy\cdot dz$$

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By symmetry, the derivation for the y and z faces is exactly the same as the x face.
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## Heat Generation and Energy Storage

- Internal Heat Generation ($Q_{gen}$):
- Let $q_g$ be heat generated per unit volume ($W/m^3$).
- $$Q_{gen} = q_g (dx\cdot dy\cdot dz)$$
- Energy Storage ($Q_{stored}$):
- Rate of change of internal energy.
- $$Q_{stored} = m \cdot C_p \cdot \frac{\partial T}{\partial t} = (\rho \cdot dx\cdot dy\cdot dz) C_p \frac{\partial T}{\partial t}$$
- where $\rho$ is density, $C_p$ is specific heat, $t$ is time.

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The generation term accounts for any internal heat source. The storage term indicates that if energy is stored, the temperature of the block rises over time.
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## General 3D Conduction Equation

- Substitute all terms back into the energy balance equation and divide by volume ($dx\cdot dy\cdot dz$):
- $$\frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) + q_g = \rho C_p \frac{\partial T}{\partial t}$$
- If thermal conductivity ($k$) is constant (isotropic material):
- $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} + \frac{q_g}{k} = \frac{\rho C_p}{k} \frac{\partial T}{\partial t}$$

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This is the general heat conduction equation in Cartesian coordinates. The term k / (rho*Cp) is a crucial property called Thermal Diffusivity, denoted by alpha.
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## Thermal Diffusivity ($\alpha$)

- Thermal Diffusivity, $\alpha = \frac{k}{\rho C_p}$
- - Units: $m^2/s$
- - Physical Meaning: Ratio of heat conducted ($k$) to heat stored ($\rho C_p$).
- - High $\alpha$: Heat propagates rapidly into the medium (e.g., metals).
- - Low $\alpha$: Heat is mostly absorbed by the material, slow propagation (e.g., wood, insulation).
- Equation becomes: $\nabla^2 T + \frac{q_g}{k} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$

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Thermal diffusivity measures how fast heat diffuses through a material. It's a vital property for transient heat transfer analysis.
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## Simplified Forms of the Equation

- 1. Steady-state ($\partial T / \partial t = 0$): Poisson's Equation
-    $$\nabla^2 T + \frac{q_g}{k} = 0$$
- 2. No heat generation ($q_g = 0$): Fourier's Equation (Transient)
-    $$\nabla^2 T = \frac{1}{\alpha} \frac{\partial T}{\partial t}$$
- 3. Steady-state & No generation: Laplace Equation
-    $$\nabla^2 T = 0$$

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Depending on the problem conditions, the general equation reduces to these well-known mathematical forms.
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## Reduction to 1D Steady State Conduction

- For 1-Dimensional (x-direction only), Steady-State, No internal heat generation:
- - 1D: $\partial^2 T / \partial y^2 = 0$, $\partial^2 T / \partial z^2 = 0$
- - Steady-state: $\partial T / \partial t = 0$
- - No generation: $q_g = 0$
- The general equation reduces to:
- $$\frac{d^2 T}{dx^2} = 0$$
- This is the governing equation for a simple plane wall.

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This simple ordinary differential equation is what we solve to find the temperature profile in a flat wall. Integrating it twice gives a linear temperature profile: T(x) = C1*x + C2.
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## Next Lecture Preview

- Topic: Heat Conduction through a Plane Wall and Composite Systems
- Key questions to ponder:
- - How do we apply the 1D steady state equation to real walls?
- - What happens when a wall is made of multiple layers of different materials?
- - How does the electrical analogy make solving composite walls easy?

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Next time, we will apply the simplified 1D equation to practical scenarios, specifically composite walls, using the thermal resistance concepts we learned earlier.
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