Consider a glass of warm 30 degree C soda is placed in the refrigerator. Where i

Question

Consider a glass of warm 30 degree C soda is placed in the refrigerator. Where is best energy being transferred as the soda cools down in the refrigerator? (b) Do a rough sketch of a graph of the temperature of a warm soda vs, time if the soda starts at 30 degree C and is placed inside a refrigerator when the temperature is 10 degree C. (c) Explain the shape of the graph, especially changes in the rate of cooling as the warm soda cools down, in terms of your observations in the lab. (d) Use a computer graphing routine to create an accurate plot of the soda's temperature is Celsius as a function of time in minute if it takes the soda 20 minutes to cool down to 20 degree C.

Explanation / Answer

So basically a refrigeration system consists of a condensor, a pump, an evaporator, and an expansion valve, also known as a throttle. What happens is that, as the refrigerant in your fridge is pumped through the evaporator, it becomes a vapor. As a vapor, it now has the capacity to absorb more energy without having the problems of (a) heat of vaporization, or (b) becoming vapor in the pump, which would break the pump (it has already passed through the pump for this cycle).

This vaporized refrigerant is pumped through the inside of the fridge, absorbing energy because it is still cooler than what's in the fridge. Then, as the refrigerant is run through the compressor, it is pushed back into the liquid state, which heats it up (work = Pv + Vp). It has absorbed energy from the fridge contents, and is, even in an idealized system, warmer than the liquid being evaporated because it has absorbed thermal energy from the inside of the fridge. This energy is transferred to air on the back of the refrigerator, which, as you might know if you've ever moved a fridge, houses a long stretch of usually copper pipe specifically for the purpose of transferring heat to the air.

So, to answer your question, heat is being transferred to the refrigerant and then to the atmosphere. What does the average temperature of the soda look like as a function of time? This can be answered by Newton's Law of Cooling:

dT/dt = h*A*(T-Tambient)

solving for T (using differential equations):

Tambient is constant, so d(T)/dt = d(T+Tambient)/dt

d(T+Tambient)/(T+Tambient) = h*A*t

Integrating over time,

T = T0 + [(T-Tambient) * exp(-h*A*t)]

This is your equation for the temperature of the soda.

T0 is the initial temperature (of the soda), 30C.

Tambient is the ambient temperature of air in the fridge, 10C.

h = the heat transfer coefficient of still air, usually about 17 W/m^2/K

A is the area of exchange, which is constant for the system.

call h*A = 'c1'

T = 30 C + (30C - 10C)*exp(-c1 * t)

The units in the exponent might seem a little weird, just remember W is J/s and J (joules) can be expressed in terms of K. h [=] W/m^2/K, A [=] m^2, and t [=] s. It might make a little more sense if you look at the equation in terms of (-h*A*t) = ln|(T-Tamb)/(T0-Tamb)|. This is known as Tlm or the log mean temp.

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