6.1. Heat Flow Fundamentals

The following topics concerning heat flow fundamentals are available:

6.1.1. Conduction and Convection

The first law of thermodynamics states that thermal energy is conserved. Specializing this to a differential control volume:

(6–1)

where:

ρ = density (input as DENS on MP command)

c = specific heat (input as C on MP command)

T = temperature (=T(x,y,z,t))

t = time

{q} = heat flux vector (output as TFX, TFY, and TFZ)

= heat generation rate per unit volume (input on BF or BFE commands)

It should be realized that the terms {L}T and {L}^T{q} may also be interpreted as T and {q}, respectively, where represents the grad operator and represents the divergence operator.

Next, Fourier's law is used to relate the heat flux vector to the thermal gradients:

(6–2)

where:

K_xx, K_yy, K_zz = conductivity in the element x, y, and z directions, respectively (input as KXX, KYY, KZZ on MP command)

Combining Equation 6–1 and Equation 6–2,

(6–3)

Expanding Equation 6–3 to its more familiar form:

(6–4)

It will be assumed that all effects are in the global Cartesian system.

Three types of boundary conditions are considered. It is presumed that these cover the entire element.

Specified temperatures acting over surface S₁:
(6–5)
where T* is the specified temperature (input on D command).
Specified heat flows acting over surface S₂:
(6–6)
where:
{n} = unit outward normal vector
q* = specified heat flow (input on SF or SFE commands)

Figure 6.1: Specified Heat Flows

Specified convection surfaces acting over surface S₃ (Newton's law of cooling):

(6–7)

where:

h_f = film coefficient (input on SF or SFE commands) Evaluated at (T_B + T_S)/2 unless otherwise specified for the element

T_B = bulk temperature of the adjacent fluid (input on SF or SFE commands)

T_S = temperature at the surface of the model

Figure 6.2: Specified Convection Surfaces

For a fluid flowing past a solid surface, the bulk temperature T_B is equal to the free stream temperature, T_FS.

For the case of bleed holes in a solid, film effectiveness (η) accounts for the coolant bleeding through the cooling holes to the external surface of the solid. The bulk temperature T_B is then a combination of the free stream temperature T_FS and the temperature of the coolant exiting the bleed hole, T_EX:

(6–8)

where 0 < η < 1.

Typically, T_EX is obtained from a FLUID116 element using the extra node option for SURF151 or SURF152.

Figure 6.3: Cooling Passages in a Solid

Note that positive specified heat flow is into the boundary (i.e., in the direction opposite of {n}), which accounts for the negative signs in Equation 6–6 and Equation 6–7.

Combining Equation 6–2 with Equation 6–6 and Equation 6–7

(6–9)

(6–10)

Premultiplying Equation 6–3 by a virtual change in temperature, integrating over the volume of the element, and combining with Equation 6–9 and Equation 6–10 with some manipulation yields:

(6–11)

where:

vol = volume of the element

δT = an allowable virtual temperature (=δT(x,y,z,t))

6.1.2. Radiation

Radiant energy exchange between neighboring surfaces of a region or between a region and its surroundings can produce large effects in the overall heat transfer problem. Though the radiation effects generally enter the heat transfer problem only through the boundary conditions, the coupling is especially strong due to nonlinear dependence of radiation on surface temperature.

Extending the Stefan-Boltzmann Law for a system of N enclosures, the energy balance for each surface in the enclosure for a gray diffuse body is given by Siegal and Howell([88](Equation 8-19)) , which relates the energy losses to the surface temperatures:

(6–12)

where:

N = number of radiating surfaces

δ_ji = Kronecker delta

ε_i = effective emissivity (input on EMIS or MP command) of surface i

F_ji = radiation view factors (see below)

A_i = area of surface i

Q_i = energy loss of surface i

σ = Stefan-Boltzmann constant (input on STEF or R command)

T_i = absolute temperature of surface i

For a system of two surfaces radiating to each other, Equation 6–12 can be simplified to give the heat transfer rate between surfaces i and j as (see Chapman([357])):

(6–13)

where:

T_i, T_j = absolute temperature at surface i and j, respectively

If A_j is much greater than A_i, Equation 6–13 reduces to:

(6–14)

where:

6.1.2.1. View Factors

The view factor, F_ij, is defined as the fraction of total radiant energy that leaves surface i which arrives directly on surface j, as shown in Figure 6.4: View Factor Calculation Terms. It can be expressed by the following equation:

Figure 6.4: View Factor Calculation Terms

(6–15)

where:

A_i,A_j = area of surface i and surface j

r = distance between differential surfaces i and j

θ_i = angle between N_i and the radius line to surface d(A_j)

θ_j = angle between N_j and the radius line to surface d(A_i)

N_i,N_j = surface normal of d(A_i) and d(A_j)

To ensure a good energy balance, it is important to satisfy both row sum and reciprocity relationships for the view factor matrix (VFSM command).

For a perfect enclosure, the row sum must satisfy the following requirement:

where F_ij are the view factor matrix values.

For a leaky enclosure, the row sum must satisfy the following requirement:

For a perfect enclosure, the following residual must be less than the specified convergence value (input as CONV on the VFSM command).

where ' indicates the new view factor values.

For a leaky enclosure, the following residual must be less than the specified convergence value (input as CONV on the VFSM command).

where F_ij is the original view factor values.

To ensure a good energy balance, the following reciprocity relationship must also be met:

where A_i is the area of the ith facet.

6.1.2.2. Radiation Usage

Four methods for analysis of radiation problems are included:

Radiation link element LINK31(LINK31 - Radiation Link). For simple problems involving radiation between two points or several pairs of points. The effective radiating surface area, the form factor and emissivity can be specified as real constants for each radiating point.
Surface effect elements - SURF151 in 2-D and SURF152 in 3-D for radiating between a surface and a point (SURF151 - 2-D Thermal Surface Effect and SURF152 - 3-D Thermal Surface Effect ). The form factor between a surface and the point can be specified as a real constant or can be calculated from the basic element orientation and the extra node location.
Radiation matrix method (Radiation Matrix Method). For more generalized radiation problems involving two or more surfaces. The method involves generating a matrix of view factors between radiating surfaces and using the matrix as a superelement in the thermal analysis.
Radiosity solver method (Radiosity Solution Method). For generalized problems in 3-D involving two or more surfaces. The method involves calculating the view factor for the flagged radiating surfaces using the hemicube method and then solving the radiosity matrix coupled with the conduction problem.