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Convection

For cable modelling, we typically assume constant thermal conductivity, see Thermal Analysis.

To consider the effect of convection (natural or forced in a gas or liquid), Newton's law of cooling is used:

$q = h A (T - T_{a})$

for air (a common cable installation medium), the convection heat transfer coefficient h, flowing at velocity v, can be obtained from:

$h = 7.371 + 6.43 v^{0.75}$

To set a convection boundary conditions, we can use a Neumann condition with q = desired convection coefficient and g = temperature of environment times the convection coefficient.

Enclosed Spaces

Convection is a complex topic, requiring a detailed understanding of heat flow, fluid flow and the behaviour of gases as they flow across surfaces. For cable sizing, to achieve this level of technical detail is not practical nor achievable in many situations. One effective technique when solving equations for enclosed spaces is the use of an effective thermal conductivity, k_e. Using an effective thermal conductivity enables the air to be treated from a conduction view, eliminating much of the complexity in implements full connectivity solutions.

For an enclosed spaced, the effective thermal conductivity is given by:

$k_{e} = k N_{u}$

The Grashof number is given by:

$G_{r δ} = \frac{g β (T_{1} - T_{2}) δ^{3}}{ν^{2}}$

with: $β = \frac{1}{T_{f}}$ and $T_{f} = \frac{T_{1} + T_{2}}{2}$

Note: T_f is in Kelvin (not °C).

Experimental results for free convection in an enclosure give:

$N_{u_{δ}} = C {(G_{r_{δ}} P_{r})}^{n} {(\frac{L}{δ})}^{m}$

Typically we can take g = 9.80665 and for air:

T_f, °C	ν, 10^-5 m².s	Pr
0	1.343	0.720
20	1.568	0.708
80	2.056	0.697
120	2.591	0.689

Note: for differing temperatures, exact factors can easily be looked up on the internet

Note: δ is the hydro-dynamic boundary layer thickness.

Values of C, n and m can be obtained from the following table using the Grashof-Prandtl number product:

Geometry	G_rδP_r	N_uδ	Pr	L/σ	C	n	m
Vertical plate	< 2,000	1.0
	6,000-200,0000		0.2-2	11-42	0.197	1/4	-1/9
	200,000-1.1x10⁷		0.5-2	11-42	0.073	1/3	-1/9
Horizontal plate	< 1,700	1.0
	1,700-7,000		0.5-2		0.059	0.4	0
	7,000-3.2x10⁵		0.2-2		0.212	1/4	0
	> 3.2x10⁵		0.5-2		0.061	1/3	0

Symbols

A           - area, m²
g - acceleration of gravity, m/s²
h                - convection heat transfer coefficient, W/m².°C
k             - thermal conductivity, W/m.°C
k_e       - effective thermal conductivity, W/m.°C
T, T₁, T₂, T_a - temperature, K
v         - velocity of air, m/s
ν - kinematic viscosity, m²/s

q - heat flow, W (or Joules/second)
$\dot{q}$ - heat generated per unit volume, W/m³

β - coefficient of thermal expansion
δ - enclosure dimension, m

G_r - Grashof number
N_u - Nusselt number
P_r - Prandtl number
R_a - Rayleigh number

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