3. So far, we have always assumed that there is no latent heat to account for. For a simple example, we can formulate a model with latent heat effects and solve it using a similarity solution. Consider a stagnant pond originally at the melting point Tm everywhere, and suppose its surface is lowered abruptly to a temperature T... :r' 3} I], (Ga) T'I[:r.r'n,t') = D for t' 3} I]. (Eb) (2111:\" 53'" . . dt' _ St 61".\" at r _ :rm. (Ge) T'[[},t'] = 1 for t' :> 0. (Ed) where St is a dimensionless parameter known as the Stefan number. Give the Stefan number in terms of p. L, c. k and Tm Ts. Next, we can show that a similarity solution I rage) = e-\"e (:3) , (7') is again possible for this problem. Substitute [7) into (Ga) and (Ed) to be satised. Find the values of or and ,"3 to allow {6a) and (Ed) to be satised. Note that these exponents are not necessarily the same as the ones we found for the heat injection problem in question 1 above [and in clam}. Show that, with these values of or and ,5. 9 satises the ordinary differential equation game 9'15) = o. (a) where E = I'ft'a. 1We still need to deal with (Eb) and (6c). Expressed in terms of the similarity variable 6,, the moving,r boundary location 33;,ft') is at :r' (f). 93m: m #5 In order for a similarity solution to hold. {m must be constant, although we do not know its value yet. Assuming that this is the case, write the boundary conditions (5b) and (So) in terms of I. 23m. 8 and 8'. Show that. with the values of tr and ,5 you htve already derived, t can be eliminated from these transformed boundary conditions, and that you obtain 1 BIKER!) 2 Earn: {93.) 9(Em) = 0- (913} (h) Solve {8] to show that are = 0mm (51) . (10) (i) The error function erf is defined as erf (x) = exp ( - 212 ) da' . Defining a function this way is not as strange as it looks: erf(x) is a function of a because r appears as a limit in the integral. Given cos(x), I could similarly define sin(x) through sin(x) = f, cos(x') da'. 'erf' is also an inbuilt function in many scientific computing packages like MATLAB. Use a change of variables in the integral to show that 9(8) = 02 + CivTerf Note that erf is defined through an integral with a fixed lower bound and the variable x (that is, the argument x of erf(x) ) as an upper bound. Conse- quently, when you integrate (10), you will have to use a definite integral with as an upper bound; remember not to use & simultaneously as the dummy variable of integration: use a different symbol. You will also need to remem ber the fundamental theorem of calculus for definite integrals, in the form f (x) - f(0) = / f'(y)dy where y is a dummy variable. If you use the fundamental theorem of calculus as just described how does 'C2' relate to O(0) ? (j) What is the value of erf(0), using the definition of erf given above? By applying boundary conditions, show that C2 = -1, Ciexp Sm FSm: 25t C2 + CivTerf 2 = 0. (11) Sketch the solution O(5). (k) The tricky part now is solving for Em. Derive a single equation for Em, not involving C1 or C2 (1) The equation you have just derived is nonlinear and can only be solved computationally. To do so, you need to specify St. For ice, we have p = 910 kg m 3, c = 2.1 x 103 J kg ! K-1, k = 2.2 Wm- K-1, L = 3.35 x 105 J kg . Consider two cases: A pond with a surface temperature of -10" C and an ocean with a surface temperature of -50 C. Find Em for both of these parameter sets from. You can for instance use MATLAB: erf is an in-built function in MATLAB, just as exp and sin are. Use the equation you derived in part k and solve it graphically - by identifying the intersection of two lines in a plot - or by any other means that you know of.) Mark Em for both cases on the plot you produced for part j(Hi) How long does a pond at 10 C surface temperature take to freeze to a depth of 2|] cm? How longr does it take for a 3 metre thick pond to freeze all the way through? To get an actual dimensional time value? you have to transform from the dimensionless variables :r', t' and 58;\" back to their dimensional equivalents I, t. and mm, using 1: = [:r]:r.', t. 2 Mr, sum 2 [x]x;1. (11) There is geological evidence that Earth went through a 'snowball' stage, in which the entire planet for almost all of it) was frozen over. If an ocean is subjected to a surface temperature of 50 0, how long does it take to freeze an ice layer of thickness i] 1 1n, ii) 100 m iii] 1000 In? [Assume the ocean is at a uniform temperature of 0 C, ignoring geothermal heat ux or heat transport by ocean currents)