Environmental Engineering Reference
In-Depth Information
Heat source (RT)
q
Heat input on surface
Control factor
φ
CH
Inner heat
conductance
control factors
Plate
FIGURE 5.69
Control factors according to temperature elevation of heating materials.
in one dimension of the thickness direction of the plate, the principal equation
(Equation 5.5) can be used for calculation of the temperature distribution in the plate:
2
2
dT
dt
dT
dx
λ
ρ
dT
dx
=
a
=
(5.5)
2
2
c
where
c
= specific heat of material
a
= thermal conductivity = λ /(ρ
c
)
ρ
= specific weight
= heat conductivity of material
It may not be easy to solve this differential equation analytically when the
boundary condition is complex. Usually, the numerical calculation of secondary
differential equation is performed after this secondary differential equation is con-
verted to a finite-difference equation by Taylor expansion.
Assuming that the temperature of a certain plane
i
in the depth direction of the
plate is
T
i
and the temperature of a plane that proceeds by the minute distance of
∆
x
into the inner direction of heat flow is
T
i
+1
, the minute distance of ∆
x
to outer
by Equation 5.5 can be converted to Equation 5.6.
t
x
(
)
(
)
j
+
1
T
=+
()
T
j
a
T
2
T
+
T
(5.6)
i
i
2
i
1
i
+
i
1
According the boundary conditions at the ends of a plate, the varying temperature
distribution in the plate can be obtained by numerically solving Equation 5.6.
For calculation of φ
CH
and λ , coiled wire is the same as plate, if we know the
φ
CH
of the surface of the coil and its heat transfer coefficient on the process of raising
the temperature of the coil. But even if coils partially touch, there are still many air
clearances. Consequently, while the heat conduction in the direction of the circle of
coiled wire can be determined only by the solid thermal conductivity of the coil,
the heat transfer in the radial direction is decided by three factors. These are the
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