Heat Diffusion Equations Lecture

Key Concepts

• Solving heat diffusion equations to determine temperature distributions.
• Procedures for approaching heat conduction problems.
• Importance of boundary conditions in solving differential equations.

Steps in Solving Heat Conduction Problems

1. Obtain the Heat Diffusion Equation: Simplify using appropriate assumptions.
2. Specify Boundary Conditions: Heat diffusion equations are second-order, needing two boundary conditions.
3. General Solution: Solve the differential equation with arbitrary constants.
4. Determine Integration Constants: Apply boundary conditions to find constants.

Example: Large Plane Wall

• Dimensions: Distance between two surfaces is 6 cm (L=6 cm), area 160 cm².
• Thermal conductivity (K): 60 W/m°C.
• Left side: Uniform heat flux (q'').
• Right side: Surface temperature 112°C.
• Objective: Determine temperature on the left surface (T1).

Assumptions and Simplifications

• Steady State: Temperature at any point does not change with time.
• One-Dimensional Heat Conduction: Heat flows in one direction (x-axis).
• Constant K: Thermal conductivity remains constant.
• No Heat Generation: Simplifies equation further.

Simplified Heat Diffusion Equation

• Original equation is simplified considering steady state, 1D heat conduction, constant K, and no heat generation.
• Two boundary conditions: Heat flux at left surface and constant temperature at right surface.

Solving the Differential Equation

1. Integrate the simplified equation once, introduce constant C1.
2. Integrate again, introduce constant C2.
3. Apply boundary conditions to solve for C1 and C2.
   • At x=0, apply left surface heat flux to find C1.
   • At x=L, apply right surface temperature to find C2.

Final Temperature Profile

• General solution: T(x) = (q''/K)x + C2, with values for q'' and K given.
• Calculating T1 when x=0 yields 117°C.

Rate of Heat Transfer

• Heat flux (q'') given, calculate the rate of heat transfer (Q) through the plane wall using Q = q'' * A or Q = K * A * ΔT / ΔX.

Extensions to Different Coordinates

Three Dimensions

• Heat flow can be considered in x, y, z directions using a small volume element and energy balance to derive the equation.

Cylindrical Coordinates

• Used for heat transfer in pipes or cylindrical objects.
• Example: Pipe with inner radius R1, outer radius R2, temperatures T1 (inner) and T2 (outer).
• Simplify using steady state, one-dimensional radial heat conduction, and no heat generation.
• Boundary conditions: Temperatures at R1 and R2.

Spherical Coordinates

• Useful for spherical objects or systems. Similar approach to cylindrical coordinates.
• Example: Spherical container with inner radius R1, outer radius R2, temperatures T1 (inner) and T2 (outer).
• Simplify and solve using boundary conditions similar to cylindrical cases.

Conclusion

• Understanding and simplifying heat diffusion equations are crucial for solving thermal problems in different geometries.
• Applying boundary conditions allows for the determination of temperature distributions and heat transfer rates.