## Appendix A-Derivation of Duct Loss Equations Using Heat Exchanger Effectiveness and Y-Delta Transformations

This derivation is for one zone only, and the nomenclature is specific to this appendix alone.

Heat transfer through the duct walls can be illustrated in the electrical analogy in Figure A-1. The first node on the left represents the temperature of the air in the duct (Td) and is connected to the temperate on the surface of the duct (Ts) by the conductance through the duct wall (Ud). The convective heat transfer coefficient (hc) connects the surface temperature to the duct zone air temperature (Ta). The radiation heat transfer coefficient (hr) connects the surface temperature to the duct zone radiant temperature (Tr).

Figure A-1: Electrical Analogy of Heat Transfer through a Duct Wall

The temperatures of the duct zone are assumed to be constant; the duct surface temperature is not. The duct surface temperature can be removed from the analysis by using a Y-Δ transform. Figure A-2 shows the result of this transformation with direct connections between the duct air temperature, the duct zone radiant and air temperatures through combined coefficients defined in Equation A-1.

Equation A-1

where

Figure A-2: Heat Transfer through a Duct Wall with Surface Temperature Removed

Using an energy balance, the rate of change of heat flow along the length () of duct must equal the heat flow through the duct wall, or

Equation A-2

where

= capacitance flow rate of the air in the duct

= temperature of air in the duct

= equivalent heat transfer coefficient (see Equation A-1)

= perimeter of duct

= temperature of air in duct zone

= equivalent heat transfer coefficient (see Equation A-1

= radiant temperature in duct zone

Regrouping by temperature terms

Equation A-3

and dividing through by the quantity (Uc P+Ur P) gives

Equation A-4

Where

Equation A-5

Let be

Equation A-6

The derivative of which is

Equation A-7

Substituting Equation A-6 and Equation A-7 into Equation A-4 gives

Equation A-8

Rearranging

Equation A-9

and integrating from entrance () to exit ()

Equation A-10

Equation A-11

Recalling the definition in Equation A-6 and replacing the product of the perimeter and length with the surface area () of the duct, and a bit of manipulation yields the following relationships

Equation A-12

Let

Equation A-13

Then

Equation A-14

Solving for the exit temperature gives

Equation A-15

The temperature change in length L of duct is

Equation A-16

This can be rewritten as

Equation A-17

Let be the sensible heat exchanger effectiveness

Equation A-18

Then the conduction loss from the duct to the duct zone can then be written as

Equation A-19