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Problem 3: Numerical Solutions to Initial Value Problems (25 Points)

Cylindrical cookies (radius: 1 cm and height 0.5 cm) are cooled on a conveyer belt after

exiting an oven. The cookies enter the conveyer belt at 200 C. The goal is for the cookies

to exit the conveyer belt at 40 C. The heat transfer coefficient at the surface of the coolie

is 25 W/m²-K. The properties of the cookies are:

C, (specific heat): 1000 J/kg-K

K (thermal conductivity) = 1 W/m-K

p (density) = 2000 kg/m³.

Your goal is to determine the time taken for the cookies to cool to 40 C using numerical

techniques./na. Write down (you don't need to solve this) the governing differential equation and

initial condition that determines the temperature variation of the cookie with time

(déjà vu!). (5 pts)

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b. Using the Euler OR 2nd order Runge Kutta methods (you may choose any method of

your choice Heun, Mid-Point, Ralston) and time step-sizes of your choosing, estimate

the time-taken for the cookie to cool down to 40 C.

Fill in your values in the table below i.e., you will be reducing your time-step size until

you get a numerically converged solution (5 points)

You don't need to use all the rows in the table if you have arrived at the numerically

"correct" answer. On the other hand, please feel free to add more rows if it has taken you

additional time-steps trials to arrive at the "correct" answer.

Time-step size

(sec/min/hrs)

Time for the

cookie to

reach 40 C

c. Choosing any time-step size and initial temperature of your choosing, show a sample

step-by-step calculation (hand calculation) that demonstrates how you used the

Euler/Runge Kutta methods to predict temperature at a future time. You only need to

show this for only one time-step. Please show the calculations clearly to enable me

to replicate your results (10 points)/nSample calculation

Initial temperature (at the beginning of the time-step):

Time-step size:

Calculation of slope or slopes (k₁ and k₂ if using Runge-Kutta Method)

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Temperature at the end of the time-step

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d. If we do not have an exact analytical solution for this scenario, how did you ensure

that you arrived at the numerically correct? (5 points)

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