Compressibility Effects

Transcription

1 Definitin f Cmpressibility All real substances are cmpressible t sme greater r lesser extent; that is, when yu squeeze r press n them, their density will change The amunt by which a substance can be cmpressed is given by a specific prperty f the substance called the cmpressibility (τ) Cnsider a small element f fluid f vlume v. The pressure exerted n the sides f the element is p. Assume the pressure is nw increased by an infinitesimal amunt dp. The vlume f the element will change by a crrespnding amunt dv; here, the vlume will decrease. Isthermal cmpressibility (τ T ) Isentrpic Cmpressibility (τ s ) 1

2 Definitin f Cmpressibility Whenever the fluid experiences a change in pressure dp, the crrespnding change in density dρ Fr a fluid flw ver an airfil, if the fluid is a liquid, where the cmpressibility τ is very small, then fr a given pressure change dp frm ne pint t anther in the flw, Equatin states that dρ will be negligibly small. In turn, we can reasnably assume that ρ is cnstant and that the flw f a liquid is incmpressible On the ther hand, if the fluid is a gas, where the cmpressibility τ is large, then fr a given pressure change dp frm ne pint t anther in the flw, Equatin states that dρ can be large. Thus, ρ is nt cnstant, and in general, the flw f a gas is a cmpressible flw The exceptin t this is the lw-speed flw f a gas; in such flws, the actual magnitude f the pressure changes thrughut the flw field is small cmpared with the pressure itself. Thus, fr a lw-speed flw, dp in Equatin is small, and even thugh τ is large, the value f dρ can be dminated by the small dp 2

3 Definitin f Cmpressibility The mst cnvenient index t gage whether a gas flw can be cnsidered incmpressible, r whether it must be treated as cmpressible, is the Mach number M Related equatins f flw? 3

4 Definitin f Ttal (Stagnatin) Cnditin Previus Understanding: Static pressure is a measure f the purely randm mtin f mlecules. The ttal (r stagnatin) pressure was defined as the pressure existing at a pint (r pints) in the flw where V = 0 In Details: Cnsider a fluid element passing thrugh a given pint in a flw where the lcal pressure, temperature, density, Mach number, and velcity are p, T, ρ, M, and V, respectively. Here, p, T, and ρ are static quantities (i.e., static pressure, static temperature, and static density, respectively Nw imagine that yu grab hld f the fluid element and adiabatically slw it dwn t zer velcity. One shuld expect (crrectly) that the values f p, T, and ρ wuld change as the fluid element is brught t rest The value f the temperature f the fluid element after it has been brught t rest adiabatically is defined as the ttal temperature, dented by T0. The crrespnding value f enthalpy is defined as the ttal enthalpy h0, where h0 = cpt0 fr a calrically perfect gas Keep in mind that we d nt actually have t bring the flw t rest in real life in rder t talk abut the ttal temperature r ttal enthalpy; rather, they are defined quantities that wuld exist at a pint in a flw if (in ur imaginatin) the fluid element passing thrugh that pint were brught t rest adiabatically 4

5 Definitin f Ttal (Stagnatin) Cnditins The energy equatin, Equatin prvides sme imprtant infrmatin abut ttal enthalpy and hence ttal temperature Assumptins which led t Equatin are that the flw is steady, adiabatic, and inviscid 5

6 Definitin f Ttal (Stagnatin) Cnditins The energy equatin, Equatin prvides sme imprtant infrmatin abut ttal enthalpy and hence ttal temperature Assumptins which led t Equatin are that the flw is steady, adiabatic, and inviscid It states that at any pint in a flw, the ttal enthalpy is given by the sum f the static enthalpy plus the kinetic energy, all per unit mass OR Ttal enthalpy is cnstant alng a streamline 6

7 Definitin f Ttal (Stagnatin) Cnditins Fr a calrically perfect gas, h0 = cpt0 Thus, the abve results als state that the ttal temperature is cnstant thrughut the steady, inviscid, adiabatic flw f a calrically perfect gas; that is 7