Fire Dynamics I

1
Fire Dynamics I

• Lecture 5
• Heat Transfer
• Conduction
• Jim Mehaffey
• 82.575 or CVG7300

2

• Heat Transfer Convection Radiation
• Outline
• Steady-state conduction
• Transient conduction
• Numerical Methods

3

• Heat Transfer by Conduction
• Heat transfer from regions of high temp to
  regions of lower temp within solids
• Important
• in ignition and spread of flame over combustible
  solids, and
• in fire resistance of combustible or
  noncombustible building elements

4

• Heat Transfer by Conduction
• Satisfies Fourier laws
• Eqn (5-1)

5

• k -Thermal Conductivity
• Materials with large k are good thermal (and
  often electrical) conductors
• Materials with small k are good thermal (and
  often electrical) insulators
• k is temperature dependent

6

• k -Thermal Conductivity (1)
• At temperatures found in Canadian climates

7

• k -Thermal Conductivity (3)
• At elevated temperatures associated with fire

8

• Steady-state Conduction
• Consider 1-D heat flow through a wall of
  thickness L
• Surfaces at temperatures Th Tc with Th gt Tc
• For example exterior
• wall in the winter

9

• Steady-state Conduction
• Fouriers Law of heat conduction (1-D)
• Eqn (5-2)
• is constant (independent of x and t)
• Integrating over thickness of wall assuming k is
  independent of temperature
• Eqn (5-3)
• Heat transfer governed by k / L

10

• Steady-state Conduction
• Integrating to depth x
• Tx Th - x / k
• or
• Tx Th - (Th - Tc) x / L Eqn (5-4)
• Tx

11

• Convection and Steady-state Conduction
• On hot side of wall, gas temperature is Th and
  convective heat transfer coefficient is hh
• On cold side of wall, gas temperature is Tc and
  convective heat transfer coefficient is hc
• Surfaces temperatures are T1 (hot side) T2
  (cold side)

12

• Convection and Steady-state Conduction
• hh (Th - T1) k / L (T1 - T2) hc(T2 -
  Tc)
• Th - T1 / hh
• or T1 - T2 L / k
• T2 - Tc / hc
• adding Th - Tc 1/hh L/k 1/hc
• or Eqn (5-5)

13

• Convection and Steady-state Conduction
• For a wall containing 3 layers with properties
  (k1, L1), (k2, L2) and (k3, L3)
• Th - Tc 1/hh L1/k1 L2/k2
  L3/k3 1/hc
• Eqn (5-6)

14

• Analogy between Electrical Thermal Resistance
• Resistors in Series
• V
• R1 R2
  R3 R4
  R5
• 1/hh L1/k1 L2/k2
  L3/k3 1/hc
• ?T

15

• Thermal Resistance
• R 1 / h (m2 K W-1) fluid to solid
• R L / k (m2 K W-1) across a solid
• In Imperial units R (ft2 F h Btu-1)
• 1.0 ft2 F h Btu-1 0.176 m2 K W-1
• For 89 mm batt insulation with (R12)
• R 12 ft2 F h Btu-1 2.10 m2 K W-1

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• Analogy between Electrical Thermal Resistance
• Resistors in Parallel
• Not a bad approximation if
• thermal resistors are thermally isolated

17

• Transient Conduction - Fire Exposure
• 1-D partial differential equation
• Eqn (5-7)

18

• Transient Conduction - Fire Exposure
• 3-D partial differential equation
• Eqn (5-8)

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• Transient Conduction - Fire Exposure
• For many applications we can assume
• k ? k (T)
• 1-D transient equation can be written
• Eqn (5-9)

20

• k -Thermal Conductivity (1)
• At temperatures found in Canadian climates

21

• k -Thermal Conductivity (3)
• At elevated temperatures associated with fire

22

• Thermally Thin Solid - Transient Heating
• ? ? Thin solid at To is
  suddenly immersed in gas current at T?
• T? T T? Assume
  convective heating on on both surfaces (Area
  2A)
• ? ? Use lumped thermal
  capacity L analysis
• Volume V A L

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• Thermally Thin Solid - Transient Heating
• Volume of solid is V A L
• Energy absorbed during time dt 2 A h (T? -T) dt
• Corresponding increase in internal energy ? c V
  dT
• ? 2 A h (T? -T) dt ? c V dT
  Eqn (5-10)
• If h T? are independent of t, integrate Eqn
  (5-10)
• (T? -T) (T? -To) exp - 2 h t / (L ? c)
  Eqn (5-11)
• Lumped thermal capacity assumption is valid
  provided
• Biot Number Bi hL/k ? 0.2

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• Transient Conduction
• Semi-finite solid
• Assume solid is initially at To
• At t 0 surface is suddenly increased to T? by
  application of heat
• Solve Eqn (5-9) subject to initial condition
  two boundary conditions

Heat
x
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• Transient Conduction
• Solution
• (T - To) (T? - To)
• (T? - To) Eqn (5-12)
• Where the error function is
• Eqn (5-13)
• Fourier number ? t / x2

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• Error Function (1)

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• Error Function

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• Depth of Penetration of Heat
• L Depth at which (T - To) is 0.5 of (T? - To)
• (T - To) 0.005 (T? - To)
• 0.005
• ? 2
• Eqn (5-14)

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• Thermally Thick Solid
• A solid is considered thermally thick
  (semi-infinite approximation is okay) if
• Thickness ?
• is a measure of depth of penetration of
  heat
• _____________________________________________
• Some researchers claim it is okay to assume a
  solid is semi-infinite if
• Thickness ?
• (T - To) 0.157 (T? - To)

30

• k -Thermal Diffusivity (3)
• At elevated temperatures associated with fire

31

• Depth of Penetration of Heat

32

• Transient Conduction
• Semi-finite solid
• Assume solid is initially at To
• For t ? 0, heat flux q (W m-2) absorbed at
  surface
• Solve Eqn (5-9) subject to initial condition
  two boundary conditions

q
x
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• Transient Conduction
• Solution
• (T - To) q -
  Eqn (5-15)
• Surface temperature (x0) is Ts
• (Ts - To) q
  Eqn (5-16)
• thermal inertia (J m-2 s1/2 K-1)

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• k -Thermal Inertia (3)
• At elevated temperatures associated with fire

35

• Transient Conduction
• q Semi-finite solid
• radiant heating
• convective
• cooling
• Assume solid and air initially at To
• For t ? 0, radiant flux q (W m-2) absorbed at
  surface which begins to undergo convective
  cooling h(Ts-To)
• Solve Eqn (5-9) subject to initial condition
  two boundary conditions

x
36

• Transient Conduction
• Solution
• Surface temperature (x0) is Ts
• Eqn (5-17)

37

38

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• Eqn (5-17)
• For small times (t ?0) (see page 5-33)
• Eqn (5-16)
• For long times (t ??)
• Eqn (5-18)
• which is bound but independent of k, ? and c

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• Transient Conduction
• T? Semi-finite solid
• Convective
• heating
• Assume solid is initially at To
• At t 0, solid is brought in contact with a gas
  T? and experiences convective heating h(T? -Ts)
• Solve Eqn (5-9) subject to initial condition
  two boundary conditions

x
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• Transient Conduction
• Solution
• Surface temperature (x0) is Ts
• Eqn (5-19)

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• For small times (t ?0)
• Eqn (5-20)
• or
• see pages 3-33 and 3-39
• For long times (t ??)
• TS T? Eqn (5-21)
• which is bound but independent of k, ? and c

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• Effect of Thermal Inertia on
• Surface Temperature Eqn (5- 19)
• Assume h 25 W m-2 K-1

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• Effect of Thermal Inertia on
• Surface Temperature Eqn (5- 19)

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• Summary - Conduction
• Steady-state conduction governed by k / L
• e.g. thermal insulation
• Penetration depth governed by
• e.g. fire resistance
• Surface temperature governed by
• e.g. ignition and flame spread

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• Numerical Methods
• Analytical solution yielding algebraic expression
  (temperature) not possible if
• Boundary conditions are nonlinear (radiation)
• Thermal properties are temperature dependent
• k k(T) ? ? (T) c c(T) Q ? 0
• Conduction is 2-D or 3-D
• Must resort to numerical (computer) solution of
  heat conduction equation

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• References
• 1. D. Drysdale, An Introduction to Fire
  Dynamics,Wiley, 1999, Chap 1
• 2. J.A. Rockett and J.A. Milke, Conduction of
  Heat in Solids Section 1 / Chapter 2, SFPE
  Handbook, 2nd Ed. (1995)
• 3. 5. T.Z. Harmathy and J.R. Mehaffey,
  Post-Flashover Compartment Fires,
  Fire and Materials Vol 7, 49-61 (1983)