4 Transient conduction
Incropera Chapter 5, Cengel Chapter 4, Welty et al. p 252-270
THIS WEEK:
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· How do we handle conduction problems in which the TEMPERATURE of the objects change with time (no longer steady state)?
· What is the difference between LUMPED parameter and DISTRIBUTED parameter systems?
Learning objectives: By the end of week 4 you should be able to:
· Calculate change in temperature over time, temperature distribution and change in internal energy for solids undergoing transient conduction, or given this information calculate other aspects of the problem, such as heat transfer coefficient or thermal properties
· Determine when temperature distribution can be assumed to be uniform
· Assess the validity of your assumptions, and explain how this affects your results.
Example 1: A Cooling Sphere, WHERE TEMPERATURE IS UNIFORM THROUGHOUT
A spherical pellet, initially at 450oC, and of uniform temperature, is cooled in air at T=20oC (h = 10Wm-2K-1). The sphere is 0.10m in diameter.
Properties of the sphere:
ρ = 3000 kgm-3
Cp = 250 Jkg-1K-1
k = 20 Wm-1K-1
Assumptions:
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Q 1a: What is the temperature of the sphere as a function of time, if the sphere is uniform in temperature?
Energy balance over the sphere:
IN – OUT + GENERATION = ACCUMULATION
Q1b. How long until the sphere reaches the safe to touch temperature of 60 oC?
Q1c What is the change to temperature after time τ?
Q1d. What is the change in temperature after 30 minutes? Is this likely to be an underestimate or overestimate? Why?
Q1e. Which equation would you use to predict the change in temperature of the sphere, if the initial temperature were 20 oC and it were heated in an oven at 450 oC?
Q1f. How much energy has been lost from the sphere after 30 minutes? If the temperature is not uniform, how will this affect your answer? What other assumptions might undermine your confidence in your answer?
If temperature is uniform throughout a solid, heating/cooling can be calculated from the rate of convection from the surface. We use
(Equation 30)
(Equation 31)
We can calculate
· Time taken to reach Tc → t(T)
· Temperature after a given time → T(t)
· Total energy lost/gained by the solid in time t
How do we test if temperature is uniform in a solid? And what happens when temperature is NOT uniform, i.e. T(0,t)≠T(r,t), 0≤r≤R?
· Introduce Dimensionless numbers.
To return to an earlier Question….. at what temperature do we evaluate thermal properties of the sphere (k, Cp, density, etc)?
The Bi Number (aka “toasted sandwich number”)
What happens when a sphere at 450 oC is cooled in air at 20 oC, but the temperature varies within the sphere?
Non-Uniform Temp Distribution Uniform Temp Distribution
Temperature, oC
Distance from centre
What affects temperature distribution?
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Resistance due to convection:
Resistance due to conduction:
Bi= =(toasted sandwich number)
Where Lc is the characteristic length scale.
Which length scale to use? It depends on the shape of the solid:
Sphere, or a long cylinder L>>ro
LT = T1 – T2
Other shapes (trapezoids, eggs etc) , where
If Bi small
· convection resistance >> conduction resistance
· temperature distribution is uniform
, as derived above
· This method is called “lumped capacitance”
If Bi large
· conduction resistance >> convection resistance
· toasted sandwich
· non-uniform temperature
· The method above is not valid, we need to do things differently
temperature gradients within solid are
negligible
Significant temperature gradients within solid
Different model required
Example 2: Revisit cooling sphere example but now include a fan: h is now 40Wm-2K-1.
The Fourier Number (Fo)
The Fourier number is a non-dimensional time, depending on thermal diffusivity α:
As for the Biot number, the length scale used in the Fourier number depends on the shape of the object:
For a sphere, or a long cylinder L>>ror
Generic shapes (trapezoids, eggs etc)
Summary: Transient Conduction
A long cylinder l>>ro Δx1 Δx2 Δx3
0.025m 0.05m 0.15m
k1 k2 k3
board pine brick
Plane wall of thickness 2L
T1 T2 T3 T4
Other shapes (trapezoids, eggs etc)
For all cases solid initially at temperature, Ti, in ambient temperature, T∞.
Neglect temperature gradients within solid
Significant temperature gradients within solid
Where ϴ = T-T∞
ϴi = Ti-T∞
If sphere, long cylinder or plane wall
Otherwise, look for other correlations, or use analytical or numerical model
= characteristic (e-folding) timescale
Use Heisler charts
: These charts provide information about the temperature profile within solids (spheres, long cylinders and flat plates) of uniform thermal conductivity, when the temperature is changing over time due to convection.
Ti is the initial temperature (assuming that the solid was initially uniform in temperature)
T∞ is the bulk temperature of the surrounding fluid
T0 is the centerline temperature
T is the temperature at distance r from the centre of sphere or cylinder of radius r0, or at distance Δx from the centerline of the plate (thickness 2Δx)
Q is the change in internal energy due to sensible heat arising from the change in temperature of the solid up to time t
Q0 is the maximum change in internal energy, i.e. the change in energy that will occur if the solid reaches the temperature of the surrounding fluid
NB: Wall thickness is 2L
Copied from Incropera and De Edition.
Example: What will be the centre temperature when the surface is at 60oC?
1. Calculate Tc = T(0,t) when T(ro,t) = 60oC
2. Calculate t when T(0,t) = Tc
at surface
A long aluminium cylinder of 5.0 cm diameter and intially at 200C is suddenly exposed to a stream of 70C. the convective heat transfer coefficient is estimated to be 525 W/m2K. caclulate the temperature at a radius of 1.25cm and the leat loss per unit length after 1 minute of exposure to air.
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