Portia and Shylock wants to serve the perfect cup of tea so as to expand their s
ID: 2083762 • Letter: P
Question
Portia and Shylock wants to serve the perfect cup of tea so as to expand their shop, dx-dTea, in Belmont (of course). Borrowing from the way espresso is made, Portia's idea is to pre-heat the tea cups by pouring boiling water (at 100 degree C) into the cup and then drain it t* seconds later, at which point the tea (at 90 degree C) is poured in immediately. Shylock is worried about wasting perfectly good boiling water and wants to know the the best time t* so as to maximize the temperature of the tea when it first sipped (assume this to be at exactly 30 seconds after it is instantaneously poured). Shylock would like to know this maximum Temperature (T_ first-sip) too. This can be modelled using state-space. The temperature of water (or tea) in the cup is one state. Then, for the cup, we can use an n-state finite element model. Then the vector x(t) sum R^n + 1 gives the temperature distribution a the fluid and the cup for a time t: where x_1 (t) is the water (or tea) temperature at time t, and x_2 (t), ..., x_n + 1 (t) is the temperatures of the elements of the cup. Then, the dynamics are given by: d/dt (x(t) - T_env) = A(x(t) - T_env) Where A sum R^(n + 1) times (n + 1) and T_env is a vector of n scalar components at a value of the env ambient temperature (T_ambinen). Now at time t = t* the liquid in the cup will change (immediately) from whatever value it has to 90 degree C (be this is the temperature of the tea being poured in). The other cause states (i e., the temperatures of the cup's elements) will not change at that instant t*. The dynamics matrix, A, is given in teamodel.m (below). [Assume for starting that T_ambinen = 22 degree C, that Temperatures are in degree C and times seconds.] How many elements, n? Please determine and justify the number of elements, n, that should be used. Optimal pre-heating time t* and Maximum T_first-sip Please determine and justify the time t* and its first-sip tea temperature As part of this justification please explain the method, submit the code, provide any supporting graphs and give final answer result (i.e., the resulting maximum first-sip tea temperature. T_first-sip for the optimal value of t*).Explanation / Answer
a.
The fundamental concept here is related to state space models.
The number of elements of the state vector will be :
cup temperature (water or tea) x1(t)
Temperature of water x2(t)
Temperature of tea x3(t)
Thus n will be 3.
As already pointed out x2 and x3 will not change at t* while x1 will change instantaneously to 90.
b.
I assume this is not expected to be solved as the details of the state transition matrix A is not provided.
Let me know your feedback on the same.