Relating the lab apartment to a full-size apartment in Beijing

Transcription

1 Achitectue Building Technology Laboatoy Sping 004 Relating the lab apatent to a full-size apatent in Beijing It is inteesting to elate what we easue in the lab to eal life. Ou lab easueents will yield ai change ates. What do these ates suggest about theal cofot in a Beijing apatent? Let s begin with the eal apatent. 1. Fist, estiate the heat load, in Watts, in the apatent. Use easonable values fo heat gains fo occupants, lights, any equipent, cooking and the sun. Please calculate you own total. Fo now, let s use 1000 W.. Let s focus on convection and assue thee is vey little heat loss by conduction. This is appopiate when thee is a sall indoo-outdoo tepeatue diffeence. Please choose a tepeatue diffeence that you would like to aintain. As an exaple, let s assue it is elatively wa outside and that we want only K diffeence that is, we will accept an indoo tepeatue that is only K wae than outdoos. 3. Calculate the aiflow V associated with you heat flow and tepeatue diffeence, by using the following equation: q = ρ C p V (T T out ) in whee ρ = density of ai, 1. kg/ 3 C p = specific heat of ai, 1000 J/kg o C V = flow ate, 3 /s Fo ou exaple, we obtain /s. 4. Expess the esult as an ai-change ate. Let s assue the volue of the apatent is Fo ou exaple, we obtain The ai-change ate is the poduct of the window aea at eithe the windwad o leewad side and the ai at the opening. The ai at the opening is NOT the sae as the fee-stea, at the sae height and in siila teain (open o uban, fo exaple) and is NOT the sae as what would be epoted by a weathe sevice. Please calculate the ai speed fo an opening aea 1

2 of you choice. If we use the nubes we have so fa and assue we have of opening, soewhee between one and two balcony doos, we then need 0.1 /s as easued at the opening. 6. Now let s look at the odel. The scale facto, n, is 10, eaning that the odel is built to a scale of 1:10. Let s use subscipts fo the eal apatent and fo the odel. Please exaine and undestand the following equations: WindowAea AiVelocit y = 3600 Volue WindowAea AiVelocit y = 3600 Volue WindowAea WindowAea = n Volue Volue = 3 n WindowAea AiVelocit y = n 3600 Volue AiVelocit y = n AiVelocit y Fo exaple, let s iagine setting up the fan and the odel such that we easue the sae ai at the opening into the living oo of ou odel as we calculated we would need to be cofotable in Beijing, 0.1 /s. Ou odel aichange ate is then ten ties what we need in eal life, o Please set up a table that deteines the ai-change ate in a eal apatent that is associated with what you easue in the lab, fo diffeent values of easued ai exchange in the lab and assued ai in eal life. Use Table 1 as a teplate. Note fo each of the fou easueent cases, you should assue two velocities in eal life. The two assued values ay diffe fo diffeent wind diections.

3 Table 1. Copaison of aiflows in odeled and full-size apatents Case nube and/o desciption No balcony at apatent entance Assued eal eal eal tepeatue diffeence Theally cofotable fo T out = 5 o C? 0 o wind diection 30 o wind diection 60 o wind diection 8. This table iplies a cetain feedo of choice in the lab. We can use whateve fan speed we want and elate the esults to what would happen in eal life. That s tue to a point: we can indeed pedict the ai-change ate in eal life. But will ou pediction be a good one? A good pediction eans that the fluid dynaics associated with ou odel will be the sae as in eal life. To ensue that is the case, we ust ensue that a non-diensional paaete known as the Reynolds Nube is the sae fo both the odel and in eal life. This in tun equies that AiVelocit y = AiVelocit y n So, if we want 0.1 /s in eal life, we will need.1 /s in the lab. Put the othe way, we can stat with what we easue in the lab and identify the ai, and tepeatue diffeence in eal life. Note that the lab is ten ties that of eal life, in ou case. This scaling facto is an issue fo wind-tunnel wok. If the scale facto is geate than ten o we want to siulate what we would have unde high wind speeds in eal life, we will need a vey poweful lab fan. Ou table-top fans ae good fo no oe than about 5 /s. Note also that if we atch Reynolds Nubes, the in ou odel will exceed that in a eal apatent by a facto of n, o 100 in this case, based on the last equation on page and the equation on this page. 3

4 Please fill in the following table, using odel velocities you easue in the lab (sae odel easueents as in Table 1): Table. Copaison of aiflows in odeled and full-size apatents that peseve fluidechanics siilitude Case nube and/o desciption No balcony 0 o wind diection 30 o wind diection 60 o wind diection at apatent entance Real that atches Reynolds Nube eal eal tepeatue diffeence Theally cofotable fo T out = 5 o C? Deivation of Reynolds Nube siilitude Consevation of oentu in two diensions fo an incopessible fluid yields the Navie-Stokes equations: u u u P u u ρ + u + v = + µ x + x t x y y v v v P v v ρ + u + v = + µ + t x y y x y whee u = in x diection v = in y diection µ = fluid viscosity 4

5 Let s conside the fist equation, unde steady flow. Futhe, let s ake this equation diensionless by defining a set of diensionless vaiables, based on a efeence length L and V (note we now use V fo and not voluetic flow), and substituting the into the Navie-Stokes equation. * x x = Lef * y y = Lef * x u = Vef * v v = Vef P * P = ρ V ef * u * * u * * u * P * ν + u u + v = + x * y * x * V ef L ef x * y * whee the kineatic viscosity ν is defined as ν = µ ρ The invese of ν V L ef ef is the Reynolds Nube, Re. Hee s the punch line. Ou odel esults should give us an accuate pediction if Re = Re We e using ai fo both, so the viscosities atch. To atch Reynolds Nubes, we insist that 5

6 = V ef L ef V ef L ef The atio of the efeence lengths is ou scale facto, n. We theefoe have V ef = V ef n 6