Heat Diffusion Equation

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Heat Diffusion Equation
Steady State Conduction
0
The above equation states Rate of change of
total energy (dE/dt) Rate of energy generated
(dEg/dt) Rate of Heat Transfer in (qin) - Rate
of Heat Transfer out (qout)
Apply this equation to a solid undergoing
conduction heat transfer as shown where
E mcpT (rV)cpT r(dxdydz)cpT
qin
x
qout
qx
qxdx
Heat diffusion equation mod 11/27/01
2
Heat Diffusion Equation (contd, page 2)
Substituting in the Energy Balance equation, we
obtain
gt
Note partial differential operator is used since
T T(x,y,z,t)
Generalized to three-dimensions
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Heat Diffusion Equation (contd, page 3)
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1-D, Steady State Conduction
Constant
Example T(x0)100CC2 T(x1 m)20CC1C2,
C1-80C T(x)100-80x (C)
T
100
20
x
5
1-D Heat, Steady State Heat Transfer Thermal
Resistance (Electrical Circuit Analogy)
Constant
q (I)
This is analogous to an electrical circuit
T2 (V2)
I (V1-V2)/R
T1 (V1)
R (R)
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Thermal Resistance - Composite Wall Heat Transfer
Use of the thermal resistance concept make the
analysis of complex geometries relatively easy,
as discussed in the example below. However,
note that the thermal heat resistance concept can
only be applied for steady state heat transfer
with no heat generation.
Example Consider a composite wall made of two
different materials
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Composite Wall Heat Transfer (contd)

• Now consider the case where we have 2 different
  fluids on either sides of the wall at
• temperatures, T?,1 and T?,2 , respectively.
• There is heat transfer by convection from the
  first fluid (on left) to material 1
• and from material 2 to the second fluid (on
  right).
• Similar to conduction resistance, we can
  determine the convection resistance,
• where Rconv 1/hA

R2L2/(k2A)
R1L1/(k1A)
Rconv,2 1/(h2A)
Rconv,1 1/(h1A)
T?,2
T1
T?,1
T
T2
where
T?,1
T?,2
This approach can be extended to much more
complex geometries (see YAC, 8-4 and 8-5)
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R value of insulations
In the US, insulation materials are often
specified in terms of their thermal resistance,
denoted as R values, where Rvalue L/k
(thickness/thermal conductivity) In the US, it
has units of (hr ft2 F)/Btu (Note 1 Btu1055
J) R-11 for wall, R-19 to R-31 for ceiling.
Q Why is the Rvalue given by L/k ? A From
previous slide, we know that the thermal
resistance to conduction is defined as the
temperature difference across the insulation by
the heat flux going through it, hence
Example The typical space inside the
residential frame wall is 3.5 in. Find the
R-value if the wall cavity is filled with
fiberglass batt. (k0.046 W/m.K0.027 Btu/h.ft.R)