Consider the simple situation where we desire to cool a couple of six-packs of soda (or whatever your favorite beverage might be) by placing the 12 cans into a cooler filled with cool water from a nearby mountain stream. Assuming the cooler is well insulated, there will be essentially no energy loss to the environment -- but energy will indeed be exchanged between the soda and water (i. e. the water will warm up as the soda cools down). Our goal here is to analyze this situation to determine how long it takes to cool your favorite beverage to some given desired temperature. Model Development
Using a sirmple lumped-parameter approximation, we can write an energy balance on the soda and water separately as
soda: mscp d/dt Ts = -hA(Ts - Tw)
water: mwcp d/dt Tw = hA(Ts - Tw)
where m, and m, are the mass of sode and water, respectively, co is the specific heat of both liquids (assumed to be the same for soda and water), and hA is an effective heat loss coefficient that takes into account both convection and conduction heat transfer. Equations (1a) and (lb) simply state that the energy lost by the soda is gained by the water which is consistent with our assumption that there is no energy transfer outside the cooler. Writter in standard form these equaions become:
d/dt Ts = -ksTs + kcTw with Ts(0) = Tso and ks = hA. mscp
d/dt Tw = kwTs - kwTw with Tw(0) = Two and kw = hA/mwcp
Assuming that the valves of k, and ky are constant, these coupled ODES define an unforced linear time-invariant (L. TI) initial value problem (IVP), which can be easily solved using either analytical er numerical techniques.
Problem-Specific Data
The physical data for the problem are given below:
Vc = cooler volume = 0.50 ft3
Vf = volume of soda = 0.15 ft3
Vw = Vc - Vw = volume of water
Two = initial temperature of water = 40 0F
Keeping It Cool
Ts = initial temperature of soda = 80 0F
Ts5 = measured temperature of the soda after 5 minutes = 68 0F
Tsd = desired temperature of the soda = 55 0F
p = density of both water and soda = 62.3 Ibm/ft3
cp = specific heat of both water and soda = 1.0 Btu/Ibm-0F
Note that the measured soda temperature at t 5 min will be used within an iterative simulation procedure to converge upon the numerical value of the hA product that gives the measured temperature valuo the data given here, one should be able to determine the T,(t) and Tw(t) profiles and, in particular, determine how long it takes to cool the soda to the desired temperature of Td S 55 °F. which then gives numerical values for the k, and kw constants. Thus, with
Analytical Solution
Write eqns. (2a) and (2b) as a matrix IVP and solve this system using the analytical Ctechniques discussed in class The result here should be a set of exact expressions for Ts(t) and Tw(t) wriuen in terms of the variables ks, kw, Tso, and Two. Be sure to only use symbolic variables at this point (no numbers yet, please), since we desire a general treatment of the subject at this point.