Math Modeling a starred variable or parameter means the variable or parameter ha
ID: 3141526 • Letter: M
Question
Math Modeling
a starred variable or parameter means the variable or parameter has dimensions. An unstarred variable or parameter is dimensionless. We are cooking spherical roast of radius R*, let B be the ball of radius R* centered at the origin. So B = {(x*, y*, z*): |x*|^2 + |y*|^2 + |z*|^2 lessthanorequalto R*^2} = {(rho*, theta, phi): rho* lessthanorequalto R*} Let u*(x*, y*, z*, t*) be the temperature at time t* at (x*, y*, z*) elementof B. The temperature on the boundary partial differential B is the oven temperature, the temperature on the inside is governed by the heat equation partial differential u*/partial differential t* = k* (partial differential^2 u*/partial differential x*^2 + partial differential^2 u*/partial differential y*^2 + partial differential^2 u*/partial differential z*^2) Non-dimensionlize the lengths with respect to R and time with respect to a natural time scale of the equation. Where is partial differential B now? Consider the non-linear system of equations dx*/dt* = -lambda * x*y* dy*/dt* = lambda* x* y* - mu* y* dz*/dt* = mu* y* Assume x*, y*, z* have dimensions of P. What are the dimensions of lambda* and mu*? What are the two natural times scales? Rescale the equations by each time scale separately to get dimensionless systems with one dimensionless parameter u. Use one of the dimensionless equations you just found above to solve the system below dy/dx = dy/dt/dx/dt dz/dx = dz/dt/dx/dt For y(x) and z(x) and use your answers to solve (what is c?) dx/dt = -cxy(x)Explanation / Answer
Ans-
Clearly, the solubility of a gas in a liquid increases as the pressure of that gas over the liquid increases. The proportionality constant H is known as the Henry’s law constant. It is temperature dependent, decreasing as the temperature increases; that is, for a fixed partial pressure, gas solubility decreases as the temperature increases. This is the reason that bubbles are seen to form on the surface of a glass of water when it is left in the sun and the reason that effervescence (bubbling) is initially greater in a hot carbonated beverage than a cold one. When the container is opened, the pressure p in the headspace falls abruptly and Henry’s law is no longer satisfied. The liquid is said to be supersaturated with gas molecules. To reestablish equilibrium or, equivalently, to satisfy Henry’s law (3.1), the dissolved gas concentration cl must decrease. The gas molecules can escape either by diffusing through the free (top) surface 145 of the liquid or by forming bubbles1 . Although the laws of thermodynamics favour the formation of gas bubbles, there is a kinetic barrier to the production of gas bubbles. To see why, we note that the gas pressure pb inside a spherical bubble of radius R is given by Laplace’s law: pb = pl + 2 R , (3.2) where pl is the pressure in the liquid and is the surface tension of the liquid in contact with the gas. Laplace’s law states that the pressure inside a bubble is greater by 2/R than that in the surrounding liquid due to the fact that the surface tension of the liquid tends to contract the bubble surface. Clearly, the gas in the bubble is in equilibrium with its concentration cl in the liquid if Henry’s law is satisfied: cl = Hpb cb, (3.3) If Henry’s law is not satisfied then the bubble either shrinks or grows. If cl < cb then the bubble shrinks as the gas in it redissolves, whereas if cl > cb then the bubble grows as gas diffuses into it from the liquid. Using Laplace’s law (3.2) to eliminate pb yields cl = H pl + 2 R . Approximating pl by pa 2 and rearranging terms yields R = 2 cl/H pa Rc, where Rc is called the critical radius. If R = Rc then the gas bubble is in equilibrium with the gas concentration in the liquid (i.e. cl = cb) and it neither shrinks nor grows. If R < Rc then cl < cb and the bubble shrinks. If R > Rc then cl > cb and the bubble grows (Figure 1). The critical radius is often given in terms of the supersaturation ratio S which is defined by S = cl Hpa 1, so that Rc = 2 paS . (3.4) If p0 is the gas pressure over the liquid in the sealed container, then the gas concentration in the liquid is initially cl = Hp0. Immediately after opening the container, the gas concentration 1What are widely called ‘bubbles’ in liquids are actually cavities: a true bubble is a region in which gas is trapped by a thin film, whereas a cavity is a gas-filled hole in a liquid. 2Strictly speaking, pl is the sum of the atmospheric pressure pa and the hydrostatic pressure lg (h z), the bubble being at a depth h z. Since pa 105 Pa and lgh . 103 Pa, the hydrostatic pressure can be neglected. 146 Figure 1: The (idealised) dissolved gas concentration close to a postcritical (R > Rc) bubble. c is the difference between the dissolved gas concentration cl in the liquid bulk and the dissolved gas concentration cb = Hpb in the bubble ‘skin’, which is a thin layer next to the bubble surface and in equilibrium with the gas pressure in the bubble. The concentration falls from cl to cb over a diffusion boundary layer of thickness N. in the liquid is still cl = Hp0 so that the initial supersaturation ratio is S = p0/pa 1. For beer ( 40 mN m1 [3]) and an initial supersaturation ratio of 2.8, which is typical for draught Guinness, the critical radius is 0.3 µm. If enough gas molecules cluster together to form an embryonic bubble with a radius greater than this critical radius, then that bubble will survive and grow. The random thermal motion of the gas molecules is the process that provides these clusters. However, in order to cluster together, the gas molecules must first push their way through the polar water molecules that are strongly attracted to each other. This places a considerable energy barrier before bubble formation. This energy barrier can be lowered by reducing the critical radius, which also increases the likelihood that enough gas molecules will randomly cluster together. To this end, the critical radius can be reduced by lowering the surface tension 3 or by increasing the supersaturation ratio S. However, calculations, verified by experiment, show that the random nucleation of bubbles, called homogeneous nucleation, requires a supersaturation ratio of several hundred to over a thousand to be observable [9, 10]. In practice, bubbles form in even weakly supersaturated liquids, such as carbonated soft drinks (S 1 [11]) or champagne (S 5 [12]). When these liquids are poured into a glass, it is easy to observe that bubbles form on the surface of the glass rather than in the bulk of the liquid. When the bubbles reach a critical size, they detach from the glass surface and rise up through the liquid. Over most of the glass surface, the bubble production is very slow, produces large ( 1 mm) bubbles, and soon stops altogether.