Consider ventilation of a well-mixed room as in Fig. P7–25. The differential equation for mass concentration in the room as a function of time is given in Prob. 7–25 and is repeated here for convenience,

There are three characteristic parameters in such a situation: L, a characteristic length scale of the room (assume L = V^1/3); V̇, the volume flow rate of fresh air into the room, and c_limit, the maximum mass concentration that is not harmful. 

(a) Using these three characteristic parameters, define dimensionless forms of all the variables in the equation.

(b) Rewrite the equation in dimensionless form, and identify any established dimensionless groups that may appear.

Data from Problem 25

An important application of fluid mechanics is the study of room ventilation. In particular, suppose there is a source S (mass per unit time) of air pollution in a room of volume V (Fig. P7–25). Examples include carbon monoxide from cigarette smoke or an unvented kerosene heater, gases like ammonia from household cleaning products, and vapors given off by evaporation of volatile organic compounds (VOCs) from an open container. We let c represent the mass concentration (mass of contaminant per unit volume of air). V̇ is the volume flow rate of fresh air entering the room. If the room air is well mixed so that the mass concentration c is uniform throughout the room, but varies with time, the differential equation for mass concentration in the room as a function of time is

where k_w is an adsorption coefficient and As is the surface area of walls, floors, furniture, etc., that adsorb some of the contaminant. Write the primary dimensions of the first three terms in the equation (including the term on the left side), and verify that those terms are dimensionally homogeneous. Then determine the dimensions of k_w. Show all your work.

Fig. P7–25

Transcribed Image Text:

dc = S – Úc – cA_kw - dt