# Cooling of bottle, beyond homework

1. Dec 3, 2015

### Vrbic

1. The problem statement, all variables and given/known data
For how long you have to let bottle in a river, if you want to cool a liquid inside from 22 °C to 12 °C, when river is 8 °C?

2. Relevant equations
Q=cm(T1-T2), Qc=l*S(T-Triver)/d *t

3. The attempt at a solution
I know how to solve it for a case when we substitute in equation Qc for conduction of heat T by average temperature of bottle from starting and finishing state. But I believe it should be possible to solve it as a differential equation for changing "T" in Qc. But I don't know how to find out dependency T(t), t- time. Or should I do it some other way?

2. Dec 3, 2015

### Staff: Mentor

Have you taken a look at Newton's Law of Cooling?

3. Dec 3, 2015

### Vrbic

Hello, thank you for response. No I didn't till now :) I'm looking for some relevant site, best is with some already solved problem, I could learn from it. Or what do you suggest?

4. Dec 3, 2015

### Staff: Mentor

I suggest a web search on "Newtons Law of Cooling example"

5. Dec 3, 2015

### Vrbic

It seems relevent T-T0=(T-T0)exp(-kt). But I gues it is solution of some differential equation.

6. Dec 3, 2015

### Staff: Mentor

It is. You can find its derivation on the web easily. It begins with the rate of temperature change being proportional to the difference in temperature between the object and its environment.

7. Dec 3, 2015

### Staff: Mentor

You have the two equations in your Relevant Equations. You just need to modify the first one a little to take into account the time dependence. Using your symbols, the transient heat balance is:
$$cm\frac{dT}{dt}=-l*S(T-T_{river})/d$$
where l is the thermal conductivity. This is the Newton cooling equation identified by gneill.

Chet