University:
Washington State University
Course:
PHYSICS 102 | General Physics
Academic year:

2024
Views:

425

Pages:

4
Author:

teespagmaxrtf

Lesson 10: Analytical Transients Last time we discussed lumped capacitance (finite-size) problems. These assumed no temperature gradients within the material. No transients problems cannot be treated as lumped because the Biot number is not sufficiently small. The simplest 1D problem is the semi-infinite body. Material is small on one side & extends infinitely in the other direction. Nothing’s truly semi-infinite, but the model is usually appropriate for short times. We will use 2 approaches: 1. Approximate Model 2. Self-similar solution ME576 Sp2013 **The Self-Similar Solution** An exact analytical solution to the semi-infinite transient problem can be obtained using a self-similar solution. * recasts problem involving 2 variables (x,t) into a problem involving only a single similarity variable η * only possible if both the PDE & BCs can be cast into a similarity form **Example Step-by-Step:** **Step 1: Define CV** **Step 2: Energy Balance** ρc(∂T/∂t) = k(∂²T/∂x²) **Step 3: Take limit as dx → 0** ρc(∂T/∂t) = k(∂²T/∂x²) **Self-Similar Equations:** T = T(η) where η = x/2√(αt) **Governing PDE:** (d²T/dη²) + 2η(dT/dη) = 0 **Boundary Conditions:** T(η = ∞) = T∞ **Solution:** T(η) = T∞ + [T(0) - T∞]erf(η) graph showing temperature distribution