Chapter_05_Sections_7_to_10 - 5 General Ventilation and the...

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5 General Ventilation and the Well-Mixed Model 5.7 Partially Mixed Conditions Sections 5.3 to 5.6 are analyses in which the concentration is uniform throughout the enclosed space, although it may vary in time, i.e. spatial uniformity but not temporal uniformity. If the ventilation volumetric flow rate (Q), source strength (S), and adsorption rate (k w ) are constant, the mass conservation equations can be integrated in closed form. If these parameters vary with time, the equations can be integrated numerically. It must be emphasized that the notion of spatial uniformity is critical to the validity of the well-mixed model and the solutions that follow from it. Unfortunately in many situations, both spatial and temporal variations in concentration occur simultaneously, i.e. the enclosed space is not well mixed. Analysis of these situations is difficult since the equations of both mass and momentum transfer have to be solved simultaneously. Numerical computational procedures are available for this and are discussed in Chapter 10. Over the years an alternative computational technique has arisen that many workers in indoor air pollution find useful. The technique employs using a scalar constant called a mixing factor (m) to modify the equations of the well-mixed model to account for non-uniform concentrations brought on by poor mixing. Consider the ventilated enclosed space with 100% recirculation shown in Figure 5.9. Other geometric configurations can be modeled in comparable fashion. Assuming well-mixed conditions and neglecting adsorption on the walls, the following expression for the contaminant can be written: S V, c(t) Q s c a Q e c(t) Q r (1- η )c(t)Q r air cleaner η fan k w A s c Q r c(t) discharge Figure 5.9 Schematic diagram of a typical ventilation system with 100% recirculation and separate make-up air. () ar dc V S Qc Qc 1 Q c Q c dt =+ − + − η r
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which reduces to a dc V S Qc Qc Q c dt =+ − − η r (5-36) To account for non-uniform mixing, mixing factor (m) is adopted, and Eq. (5-36) can be rewritten as a dc VS m Q c m Q c m Q dt −η r c (5-37) Eq. (5-37) can be written in the standard form of Eq. (5-7) as usual, with ( ) r a mQ Q Sm Q c A B VV + == If m, S, c a , Q, Q r , and η are constants, the ODE can be solved in closed analytical form using Eqs. (5-10) and (5-11), ( ) [] ( ) ss r ss cc t m QQ exp At exp t ( 0 ) V −+ =−=− η (5-1) where () a ss r Q c B c AmQ Q + (5-2) Esmen (1978) states that values of m are normally 1/3 to 1/10 for small rooms and possibly less for large spaces. Table 5.1 contains values of m referenced by Repace and Lowery (1980). If m is less than unity, the concept of mixing factor suggests that a fraction of each flow, mQ and mQ r , is well mixed while another fraction, (1 - m)Q and (1 - m)Q r , bypasses the enclosure. Consequently - m = 1 implies well-mixed model and concentration that is spatially uniform - m < 1 implies nonuniform mixing and spatial variations in concentration The parameter “m” is a discount rate or handicap factor. It implies that the enclosed space is a well-
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Chapter_05_Sections_7_to_10 - 5 General Ventilation and the...

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