# Gray surfaces

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Because wavelength-dependent properties are often not available, the net radiation method is quite often applied under the further assumption that all surfaces in the enclosure are gray. In that case, the three basic equations can be integrated over wavelength to become

${J_k} = {\varepsilon _k}{E_{b,k}} + {\rho _k}{G_k} = {\varepsilon _k}\sigma T_k^4 + \left( {1 - {\varepsilon _k}} \right){G_k}\qquad \qquad(1)$
${q''_k} = {J_k} - {G_k}\qquad \qquad(2)$
${G_k} = \sum\limits_{j = 1}^N {{J_j}} {F_{k - j}}\qquad \qquad(3)$

The equations can be written for each of the N surfaces in an enclosure, resulting in 3N equations in 3N unknowns (Jk,Gk and either qk or Tk for each surface). Before proceeding to some examples, the method can be simplified still further. In most radiative transfer problems, the explicit value of G is not needed, so G can be algebraically eliminated from the equation set. The Gk can be found from eq. 2 and substituted into eqs. 1 and (3), giving

${J_k} = {\varepsilon _k}\sigma T_k^4 + \left( {1 - {\varepsilon _k}} \right)\left( {{J_k} - {{q''}_k}} \right)\qquad \qquad(4a)$

Or

${J_k} = \sigma T_k^4 - \frac{{\left( {1 - {\varepsilon _k}} \right)}}{{{\varepsilon _k}}}{q''_k}\qquad \qquad(4b)$

And

${J_k} = {q''_k} + \sum\limits_{j = 1}^N {{J_j}} {F_{k - j}}\qquad \qquad(5)$

Using eqs. (4b) and (5) for each of the N surfaces reduces the set to 2N equations in 2N unknowns. The radiosity is sometimes needed in an engineering solution, because it is the quantity that is observed by radiometers or radiation pyrometers viewing a surface in an enclosure. If the radiosity is not required, then it can also be eliminated from the equation set by substituting eq. (14b) into eq. (5) to give

$\sigma T_k^4 - \frac{1}{{{\varepsilon _k}}}{q''_k} = \sum\limits_{j = 1}^N {\left[ {\sigma T_j^4 - \frac{{\left( {1 - {\varepsilon _j}} \right)}}{{{\varepsilon _j}}}{{q''}_j}} \right]} {F_{k - j}}\qquad \qquad(6)$

To place the temperatures on one side of the equation, eq. (6) is rearranged to the form

$\sum\limits_{j = 1}^N {{F_{k - j}}\sigma \left( {T_k^4 - T_j^4} \right)} = \sum\limits_{j = 1}^N {\left[ {\frac{{{\delta _{kj}}}}{{{\varepsilon _j}}} - {F_{k - j}}\frac{{\left( {1 - {\varepsilon _j}} \right)}}{{{\varepsilon _j}}}} \right]} {q''_j}\qquad \qquad(7)$

where δkj = 1 if k = j and δkj = 0 otherwise. Equation (7) is written for each surface k in an enclosure, resulting in N equations in N unknowns. One consequence of the form of eq. (7) is that the temperature of at least one surface in the enclosure must be specified; if only q" is specified for each surface, the solution for Tk becomes indeterminate.

For pure radiative transfer (no conduction or convection present), eq. (7) results in a set of linear equations with unknowns T4 or q" for each surface; any standard method for solving a set of linear equations can be used to determine the unknowns. The matrix of coefficients that results from the formulation may be fully populated if every surface in the enclosure can exchange energy with every other element.

The value of q" is the rate of energy per unit area transferred by radiation; the sign convention chosen is that q" is the heat flux added to the surface to balance the radiative transfer.

Applying eq. (7) to the case of infinitely long concentric circular cylinders results in another useful relation

${q''_1} = \frac{{\sigma (T_1^4 - T_2^4)}}{{\frac{1}{{{\varepsilon _1}}} + \left( {\frac{{{D_1}}}{{{D_2}}}} \right)\left( {\frac{1}{{{\varepsilon _2}}} - 1} \right)}}\qquad \qquad(8)$

Equation (7) can be solved for a larger number of surfaces as follows. Taking the case of the temperatures of all N surfaces known, the equation becomes

$\sum\limits_{j = 1}^N {\left[ {\frac{{{\delta _{kj}}}}{{{\varepsilon _j}}} - {F_{k - j}}\frac{{\left( {1 - {\varepsilon _j}} \right)}}{{{\varepsilon _j}}}} \right]} {q''_j} = \sum\limits_{j = 1}^N {{F_{k - j}}\sigma \left( {T_k^4 - T_j^4} \right)} \qquad \qquad(9)$

Let

$\sum\limits_{j = 1}^N {\left[ {\frac{{{\delta _{kj}}}}{{{\varepsilon _j}}} - {F_{k - j}}\frac{{\left( {1 - {\varepsilon _j}} \right)}}{{{\varepsilon _j}}}} \right]} {q''_j} = {a_{kj}}{q''_j}{\rm{; }}\sum\limits_{j = 1}^N {{F_{k - j}}\sigma \left( {T_k^4 - T_j^4} \right)} = {C_k}\qquad \qquad(10)$
.

Then ${a_{kj}} = \left[ {\frac{{{\delta _{kj}}}}{{{\varepsilon _j}}} - {F_{k - j}}\frac{{\left( {1 - {\varepsilon _j}} \right)}}{{{\varepsilon _j}}}} \right]$. For an enclosure of N surfaces, the set of equations becomes

$\begin{array}{l} {a_{11}}{{q''}_1} + {a_{12}}{{q''}_2} + ... + {a_{1j}}{{q''}_j} + .......{a_{1N}}{{q''}_N} = {C_1} \\ {a_{21}}{{q''}_1} + {a_{22}}{{q''}_2} + ... + {a_{2j}}{{q''}_j} + .......{a_{2N}}{{q''}_N} = {C_2} \\ {\rm{ }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{.}} \\ {\rm{ }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{.}} \\ {\rm{ }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{.}} \\ {a_{N1}}{{q''}_1} + {a_{N2}}{{q''}_2} + ... + {a_{Nj}}{{q''}_j} + .......{a_{NN}}{{q''}_N} = {C_N} \\ \end{array}\qquad \qquad(11)$

In matrix form, this is Aq" = C, where the Amatrix is

A=
$\begin{array}{l} {a_{11}} + {a_{12}} + ... + {a_{1j}} + .......{a_{1N}} \\ {a_{21}} + {a_{22}} + ... + {a_{2j}} + .......{a_{2N}} \\ {\rm{ }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }} \\ {\rm{ }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }} \\ {\rm{ }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }}{\rm{. }} \\ {a_{N1}} + {a_{N2}} + ... + {a_{Nj}} + .......{a_{NN}} \\ \end{array}\qquad \qquad(12)$

The solution for q"j is found using standard matrix inversion routines to find the inverse of matrix A, denoted by A − 1, and the solution for q" is

$q’’ = {A^{-1}}C \qquad \qquad (13)$

Radiation shields An important application of radiative transfer between infinite parallel plates and between infinitely long concentric circular cylinders is in the construction of multilayer insulation. Multiple closely-spaced sheets of low-emissivity material are used to make highly insulating light-weight blankets for many uses, including thermal blankets for campers and fire protection blankets as an emergency shelter for forest fire fighters.

Consider N shields of very thin material coated on each side with a low emissivity coating of emissivity ε as shown in Fig. 2. Let the upper surface 1 be at temperature T1, and the lower surface at T2. The top and bottom plates have the same emissivity as the shields. Treating the radiative transfer between any two layers as a series resistance heat transfer problem, the result of radiative heat transfer between two surfaces can be placed in the form

$q'' = \frac{{\sigma \left( {T_j^4 - T_{j + 1}^4} \right)}}{{\left[ {\frac{1}{\varepsilon } + \frac{1}{\varepsilon } - 1} \right]}} = \frac{{\sigma \left( {T_j^4 - T_{j + 1}^4} \right)}}{{\left[ {\frac{2}{\varepsilon } - 1} \right]}} = \frac{{\sigma \left( {T_j^4 - T_{j + 1}^4} \right)}}{R}\qquad \qquad(14)$

where R = [(2/ε)-1] is the radiative resistance. For N = 1, there will be two gaps between the plates, so

${q''_{N = 1}} = \frac{{\sigma \left( {T_1^4 - T_2^4} \right)}}{{2R}}$

or, in general for N shields between the two bounding plates,

${q''_{{\rm{N shields}}}} = \frac{{\sigma \left( {T_1^4 - T_2^4} \right)}}{{(N + 1)R}}\qquad \qquad(15)$

The ratio of heat transfer with N shields in place to that with no shields is

Figure 3: Multilayer shield development for the James Webb space telescope. The full-scale sunshield is about 22 by 12 m.(Courtesy NASA)
$\frac{{{{q''}_{{\rm{N shields}}}}}}{{{{q''}_{{\rm{no shields}}}}}} = \frac{1}{{(N + 1)}}\qquad \qquad(16)$

One shield cuts the radiative transfer in half, and 10 shield layers drops the radiative transfer to 9 percent of the unshielded value.

Spacecraft use this type of insulation blanket, since in the vacuum of space there is no gas between the shield layers and conduction and convection heat transfer that may be present between the layers on Earth are suppressed. (Actually, thin blankets may use a dimpled surface on each layer to provide spacing between layers, and the small dimples provide a minor conduction path even in space.) Insulation blankets to minimize temperature swings of electronics and other temperature-sensitive packages exposed to sun-shade orbits usually have 5 to 30 layers. The multilayer blankets are also used to shield cryogenic vessels, where environmental temperatures near 300K can otherwise provide a significant radiation flux to cold cryogenic tank surfaces.

The NASA James Webb space telescope will view objects mostly in the infrared spectrum (0.6 to 28 εm), and so must be maintained at a very low temperature (below 50K) to minimize IR signal interference from local radiation sources. A large five-layer radiation shield is used to minimize solar-induced temperature gradients and any resulting thermal distortions of the telescope mirrors as well as to maintain the required low temperature. The shield layers are made of Kapton with an aluminum/doped-silicon coating on the front face (for good radiative properties as well as micrometeoroid protection) and a thin aluminum layer on the other surfaces. The orbit of the telescope will allow the shield to block radiation from the Sun, Moon, and the Earth. A photo of the development shield is shown in Fig. 3.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.