Simplest Cases
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Rate of heat conducted through solid of thermal conductivity, k, assuming steady state, 1-d conduction.
Rate of heat transferred from surface, TS, to adjacent fluid, T∞, through convection.
Radiation emitted from a surface, Ts, with emissivity, ε
2 STEADY STATE 1-D CONDUCTION
Incropera p 73-110, Cengel p75-98, Welty et al. p 217-222
This week:
Where does the steady-state 1-d conduction equation come from?
Under what conditions can and can’t it be used?
What happens if:
· Thermal conductivity is not uniform?
· Conduction and convection occur in series?
· Conduction through a sphere or cylinder rather than slab?
Learning objectives, tasks, readings, Learn ChemE videos
1. Appreciate that analytical equations in Heat Transfer can all be derived from the general heat equation:
a. Learn ChemE videos “The Heat Equation”
b. Apply simplifying assumptions and understand how they help you to solve problems.
c. Note – derivations not required in this unit
2. Use the concept of “resistances in series” to determine:
a. Overall heat transfer coefficient, U, for composite walls
b. Heat transfer rate through composite walls
c. Both a and b in plane, cylindrical, and spherical walls
By the end of the week you should be able to:
· Calculate q, R’s, T’s from composite wall problems in either plane, cylindrical or spherical co-ordinate systems.
Derivation of General Heat Equation
Energy balance over a control volume
IN – OUT + GENERATION = ACCUMULATION
Integrate over control volume
Equation H-1
How to solve?
Specify initial condition and boundary conditions, e.g.
· Isothermal surface
· Constant heat flux to surface
· Insulated surface
· Convective surface condition
OR, Simplify!
Common simplifying assumptions:
No internal generation (no reactions):
Equation H-2
Steady state:
Equation H-3
No convection:
Equation H-4
DERIVATION: CONDUCTION THROUGH A SOLID WALL
Incropera p. 74-79, Cengel p. 135-162, Welty et al. p 224-233
Determine heat transfer through the wall under the same conditions as last week, but now consider a wall made of plasterboard (2 cm), pine (4 cm) and brick (14 cm):
Assumptions:
· _____________________________________________
· _____________________________________________
· _____________________________________________
· _____________________________________________
Rate of heat conducted through plasterboard:
(1) Equation H-5
Rate of heat conducted through pine:
(2) Equation H-6
Rate of heat conducted through brick:
(3) Equation H-7
Solve simultaneously
Equation H-8
DERIVATION: CONDUCTION THROUGH A SOLID WALL – THE WALL CAN ALSO BE MODELLED AS THREE RESISTANCES IN SERIES:
Composite walls have overall heat transfer coefficient, U:
Equation H-9
Rate=_________________________
Equation H-10
Equation H-11
Cf Equation H-8
SUMMARY: RESISTANCES IN SERIES and Overall Heat Transfer Coefficients
Case 1 – area constant across all layers
· Calculate Rtotal, including units to avoid errors.
· Comment – which resistance dominates?
· Consider how this is likely to affect temperature profile.
Calculate [U]=Wm-2K-1.
or for final answer.
cylindrical co-ordinates
Example: 1-d conduction through a CYLINDRICAL shell
Area for heat transfer at radius r1
Assumptions:
· _____________________________________________
· _____________________________________________
· _____________________________________________
Now consider resistances in series in cylindrical coordinates.
area is not constant across the different layers, hence use ua rather than u
Since area varies across each layer, it must be included in resistance:
· Calculate Rtotal, including units to avoid errors.
· Comment – which resistance dominates?
· Consider how this is likely to affect temperature profile.
Calculate [UA]=WK-1.
or for final answer.
Summary – 1-d, Steady State Conduction
PLANE WALL: Heat conducted across a thickness ∆x
1-d conduction (polar co-ordinates)
Heat conducted through a hollow cylinder
Equation H-12
Heat conducted through a hollow sphere
Equation H-13
Resistances added in series as before but area varies with radius so define
Equation H-14
Where Equation H-15
For example, for a cylinder
Equation H-16
If Δr<