1.6 Distribution of SW and LW Radiation inside the Zone

The radiant model used in CSE is based on the “MRT Network Method” developed by Joe Carroll (see Carroll 1980 & 1981, and Carroll & Clinton 1980 & 1982). It was chosen because it doesn’t require standard engineering view factors to be calculated, and yet gives a relatively accurate radiant heat distribution for typical building enclosures (see Carroll 1981).

It is an approximate model that simplifies the “exact” network (see Appendix C) by using a mean radiant temperature node, Tr, that act as a clearinghouse for the radiation heat exchange between surfaces, much as does the single air temperature node for the simple convective heat transfer models. For n surfaces this reduces the number of circuit elements from (n-1)! in the exact case, to n with the Carroll model.

For black surfaces the radiant network is shown in Figure 8. For n surfaces, T_r floats at the conductance, , A_i F_i weighted average surface temperature:

The factor F_i, in the radiant conductance between the T_i surface node and the T_r node

i in the temperature T_r. The F_i factors are obtained from the set of n nonlinear equations for n surfaces:

Given the surface areas, these equations are easily solved at the beginning of the simulation by successive substitution, starting with all F_i=1. This converges for realistic enclosures, but won’t necessarily converge for enclosures having only two or three surfaces, particularly if there are large area disparities.

F_i is always larger than 1 because it’s role is to raise the conductance between T_r and T_i to compensate for the potential difference |T_r - T_i |being smaller than it would be had T_i not been part of the conductance weighted average

i values can be seen to be close to 1, since

Using a Y-Δ transformation, the Figure 8 circuit can put in the form of the exact solution network of Figure C-1 in Appendix C, showing the implicit view factors

Thus the implicit view factors are independent of the relative spacial disposition of the surfaces, and almost directly proportional to the surface area A_j of the viewed by surface i. Also, without special adjustments (see Carroll (1980a)), all surfaces see each other, so coplanar surfaces (a window and the wall it is in) radiate to each other.

Equation 79 is exact (i.e., gives same answers as the Appendix C model) for cubical rooms; for which

Equation 78 gives F_i=1.20. Substituting this into Equation 80 gives the implied F_ij=0.2. This is the correct Fij for cubes using view-factor equations Howell(1982). It is likely accurate for all of the regular polyhedra.

Carroll’s model handles gray surfaces, with emissivities, ε_i by adding the Oppenheim surface conductance

in series with the conductances h_b A_i F_i . As shown in Figure 9, the conductance between T_i and T_r becomes hb Ai Fi where the Fi' terms are:

where for grey surfaces T_r is the " hb Ai Fi" weighted average surface temperature given by:

Similar to Equation 77 for a black enclosure, Equation 83 shows that T_r for grey surfaces is the conductance, A_i F_i, weighted average surface temperatures.

The role of F_i hasn’t changed, but since the conductance A_i F_i is now connected to the radiosity node rather than the surface node,

i F_i-weighted average radiosity of the surfaces, rather than the A_iF_i -weighted average emissive power of the surfaces as in the black enclosure case.

This completes the description of the basic Carroll model. The principle inputs are the interior surface areas in the zone, the emissivities of these surfaces, and the typical volume to surface area ratio of the zone (see Section 2.6.1.3). All of the interior surfaces, including ducts, windows, and interior walls, are assumed to exchange heat between each other as diffusely radiating gray body surfaces.

Longwave radiant internal gains can be added, in Btu/hr, to the radiant node Tr. This distributes the gains in proportion to the conductance

Using a Y-Δ transformation, the radiant network of Figure 9 can be converted to the Appendix C, Figure C-3 circuit form, eliciting the Fij' interchange factors implicit in Carroll’s algorithm. Similar in form to Equation 80,

The total net heat transfer from surface i (i.e., the radiosity minus the irradiation for the un-linearized circuit) is given by summing Equation 85 for all the surfaces seen by surface i:

1.6.1.2 Accuracy of Carroll model

The Carroll model of Figure 9 is exact for cubical enclosures with arbitrary surface emissivities. It is surprisingly accurate for a wide variety of shapes, such as hip roof attics and geodesic domes.

Carroll (1981) compared his model, and other simplified models, with the exact solution for the enclosure shown in Figure 10. Half the south wall and half of the west wall are glass with ε = 0.84, and the rest of the interior surfaces have ε = 0.9.

The first law requires that the sum of the net radiation emitted by each of the surfaces, plus any internal gain source of long-wave radiation, must equal zero. That is,

Due to their fixed conductance circuits, both the Carroll method and the Walton(1983) method are inherently free of heat balance errors. Carroll found BLAST and NBSLD algorithms to have rms heat balance errors of 9.8% (12%) and 1.7%(3.4%) for the Figure 10 enclosure.

For a given enclosure, these are errors in an individual surfaces net heat flow, q_i, compared to the exact method. For Carroll’s method, this finds the error in q_i, determined from the

Carroll found the % rms error in the q_i, values for a given enclosure in two different ways.

The first method, Equation 87, gives the rms error of q_i, for each surface divided by the rms of the n net heat transfers from each surface:

The second method, Equation 88, gives the rms of the percentage error in q_i f each surfaces. This method increases the weight of smaller surfaces such as windows.

For the enclosure of Figure 10, Carroll found his method gives Err = 0.11% for the first method and 0.19% for the second method.

These results are shown in Table 4, along with the results for other shape enclosures, and the errors determined by Carroll using the radiant interchange algorithms of Walton (1980) and NBSLD and BLAST simplified models.

Table 4: Err = %rms Error in q_i from Equation 87 and Equation 88 in Parenthesis

	Figure 10 room 2.5:4:6 ε = 0.9 (.84 wdws)	Corridor 10:1:1 ε = 0.9	Warehouse 10:10:1 ε = 0.9
Carroll	0.11 (0.19)	0.06(0.05)	0.07 (0.04)
Walton(1980)	1.9 (1.30)	0.6 (0.6)	4.4 (3.0)
NBSLD, BLAST	3.2 (2.1)	3.2 (2.6)	7.5 (4.4)

Errors in an individual surface’s distribution of heat transfer to other surfaces

These are errors in q_ij, the heat exchanged between surfaces i and j (both directly and by reflections from other surfaces), relative to the exact total net heat transfer from surface i given by Figure C-3 of Appendix C.

By the first method, for each surface i, the rms of the error, Δq_ij, in heat exchange to each of the n-1 other j surfaces is obtained. Then the rms of these n rms error values is obtained, giving a representative distribution error for the enclosure. Dividing this by the rms value of the exact net surface heat transfers, q_i, of all the surfaces gives the final distribution error in percent:

By the second method, for each surface i, the rms of the percentage error in heat exchange q_ij relative to the exact net heat transfer from that surface, q ̅_i,is obtained.

For the Figure 10 room, Carroll’s model gives errors of 2.1% and 3.9% for methods 1 and 2 respectively. Walton’s model has corresponding errors of 2.4% and 3.7%. Equation 91 was used for the results in parenthesis.

	Figure 10 room 2.5:4:6 ε = 0.9 (.84 wdws)	Corridor 1:10:1 ε = 0.9	Warehouse 1:10:10 ε = 0.9
Carroll	2.1 (3.9)	3.3 (2.8)	0.6 (1.9)
Walton(1980 )	2.4 (3.7)	3.3 (2.8)	2.8 (3.0)
BLAST	2.8 (4.4)	3.4 (4.4)	3.4 (15)
NBSLD	1.7 (3.5)	1 (0.83)	3.3(1.9)

Carroll’s model is seen to give very respectable results, despite giving no special treatment to coplanar surfaces.

1.6.1.3 Air absorption

The Carroll model also accounts for the absorption of long-wave radiation in the air, so that the air and mrt nodes are thermally coupled to each other as well as to the interior surfaces. Carroll (1980a) gives an air emissivity by the following dimensional empirical equation that is based on Hottel data from McAdams(1954):

ε_s= the area-weighted average long-wave emissivity for room surfaces, excluding air.

Following a heuristic argument Carroll assigns an effective area Aa to the air that is the product of εa and the sum of all of the zone surface areas, as if the absorbing part of the air were consolidated into a surface of area

Using this area, the value of F_a for this 'surface' can be calculated along with the other F_i by Equation 80. The value of the conductance between the air and radiant nodes in Figure 11 is given by:

Suppose one of the interior surfaces of total area A_i is composed of N_i identical flat sub-surfaces, each at the same temperature, and similar views to each other, like the facets of a geodesic dome. The F_i values would be the same if each facet is treated as a separate surface. To avoid redundant solutions to Equation 80, it is easy to show that A_i can be treated as one surface in Equation 6-4 if iN_i is introduced into Equation 80 as follows:

The facet feature is utilized in the simulation to represent attic truss surfaces.

This routine was used in the development code for this program. It is not currently implemented in CSE, being replaced by a simplified but similar routine.

The short wave radiation (solar insolation from hourly input) transmitted by each window can, at the users discretion, be all distributed diffusely inside the zone, or some of the insolation from each window can be specifically targeted to be incident on any number of surfaces, with the remaining untargeted radiation, if any, from that window, distributed diffusely. The insolation incident on any surface can be absorbed, reflected, and/or transmitted, depending on the surface properties inputted for that surface. The radiation that is reflected from the surfaces is distributed diffusely, to be reflected and absorbed by other surfaces ad infinitum.

Since some of the inside surfaces will be the inside surface of exterior windows, then some of the solar radiation admitted to the building will be either lost out the windows or absorbed or reflected by the widows.

1.6.1.4 Radiation removed at each surface of a zone by a single source of targeted insolation

Assume a spherical zone with total insolation S(Btu/hr) admitted into the zone through one window. Assume that the portion

2*Btu/(hr-ft²))

Also incident on surface i will be the irradiation G_i (Btu/hr-ft²) from other surfaces that have reflected a portion of the radiation they have received. We distinguish between the Q_i incident on the surface directly from the window, and the irradiation G_iwhich is composed of radiation reflected to i from all the surfaces, and that reflected by windows. All radiation (including Incident beam) is assumed to be reflected diffusely.

Each surface i will also reflect short-wave radiation, with a radiosity J_i [Btu/hr-sf].

The derivation below determines the equations to obtain J_iand G_i for known Q_i values, for all the surfaces of the sphere, i = 1 to n.

For spherical geometry, the view factor is F_ik=a_k/a_S, where , a_S=∑a_k so G_ican be written

The right hand side is the area-weighted average radiosity, showing that G_i is independent of i,

Next a separate relationship between J_iand G_i is obtained, G_i eliminated and J_i solved for explicitly:

The radiosity of surface i is composed of the reflected part of both the irradiation and the targeted solar

Since by Equation 98 G_i is independent of i, then Equation 99 shows that the radiosity of any surface i is related to the radiosity of any surface k by the relationship

The first term in Equation 104 is from the absorption and/or transmission of radiation that reached and is absorbed by surface i after having been reflected, ad infinitum, by the interior surfaces. The second term is from the absorption of the "initially" incident insolation Q_i on surface i.

If none of the insolation is specifically targeted, and instead S is assumed to be distributed isotropically then Q_i is the same for each surface:

Substituting this into Equation 104 gives Qnet_i for isotropically distributed insolation:

1.6.1.5 Radiation removed at each surface of a zone by multiple window sources of targeted insolation

The targeting can be different for each window. Adding an additional subscript "j" to Equation 104 allows it to represent the energy removal for each surface separately for each window j. That is, Equation 104 becomes Equation 107, the rate of energy removal at each surface due to insolation S_j, that is distributed according to the assigned targeted values

The targeting fractions H_jk to be user input, are defined as the fraction of insolation from window j that is incident on surface k:

Replacing the spherical surfaces a_i in Equation 107 by a_i=A_i F_i, and substituting Equation 109 into Equation 110 gives the targeted gain equation used in the CZM code:

If ∑_kH_jk<1 then it is assumed that the remaining insolation S_j(1-∑_kH_jk) is distributed isotropically. From Equation 105 it is

Changing Equation 112 to utilize zone areas, ai=Ai Fi and substituting Equation 112 into Equation 113 gives the amount of the diffuse part of the insolation from each window j that is

Note that no distinction has been made between surfaces that are opaque like walls, and partially transparent window surfaces. They are treated equally. The difference is that the energy removed by an opaque wall is absorbed into the wall, whereas that removed by the window surfaces is partly transmitted back out the window, and partly absorbed at the window inside surface. The CZM development code lets the user specify a fraction of the radiation that is absorbed in the room-side surface of the window, which slightly heats the window and thus the zone.

Adding Equation 111 and Equation 114 gives the total effective absorptivity of surface i from the insolation admitted through window j:

Summing this over all windows gives the total SW radiation absorbed and/or transmitted by surface i as: