Unsteady State Heat Conduction

Scope

• One dimensional conduction• Transient condition (Unsteady state)

– Temperature as a function of time and position

• Analysis– Lumped Systems– Average temperature analysis – Transient heat conduction in infinite and semi-

infinite solids, Convective boundary conditions

One dimensional transient heat conduction

• Thermal diffusivity• T(x,t), 2 BC and 1 IC

– Analytical method using Method of Separation of Variables

– Numerically by Finite difference or Finite element method

– By Charts and Graphs

• Lumped analysis: T(t)

Lumped System Analysis

• Bodies behave like a “lump” whose interior temp. remains uniform at all time during the heat transfer process

• Example– A hot copper ball

coming from an oven

Validity of Lumped Analysis

• Assumed to cases where temperature gradient within the solid is small

• Occurs if the thermal resistance within the solid is very small compared to the external thermal resistance

• Solids with very high thermal diffusivity

• Applicable if Biot Number is less than 0.1

• s is the thermal time constant– A product of the resistance to convection heat

transfer and lumped thermal capacitance

Heat transferred

0

[ ]t

q t hA T t T

Q q t dt

Derive Q for body which lump system analysis is valid…

Biot number and Characteristic length

Compute the Biot Number

Compute the Biot Number

Derive the following equation:

Fourier Number (dimensionless “time”)

Fourier Number (Fo)

Time of death analysis

Ans. 12.2 hr

What if the BIOT number is greater than 0.1?

Average Temperature Analysis

(McCabe and Smith, 5th ed)

slab

Sphere

Infinite long cylinder

Sphere

Heat transferred using the average temperature

T p p ave aQ mc T mc T T

Slab

Infinite long cylinder

Sphere

Temperature (position,time)Local Temperature Analysis

Geometric and thermal symmetry

Transient Temperature ChartsLocal temperature is a function of position and time (Heisler and Grober charts). Check for such charts in ChE Handbook/Unit Ops book…

Always check the parameters to be used in the chart …

ChartSample Heisler charts

Analysis for semi-infinite solids

• When solids are heated and cooled such that the temperature changes in the solids are found in the region near one surface only

• e.g., thick wall, earth surface

Complementary error function

( ) 1 ( )erf x erfc x

(McCabe and Smith, 5th ed)

Penetration distance

• The distance where the temperature change has reached about 1% of initial change in surface temp.

Heat transferred

Minimum burial depth of water pipes to avoid freezing

Answer: about 0.80 m

Determine the penetration distance for this condition…. (about 4 m?)

Transient heat conduction in multi-dimensional system

Superposition approach (Product solution)

The solution for a multi-dimensional geometry is the product of the solutions of the one dimensional geometries whose intersection is the multi-dimensional body.

Total transient heat transfer

Exercises

Answer: about 15 minutes

Exercises