# Heat Tranfer Problem with Varying Temperature

• tejas710
In summary, the task at hand is to get coffee for a research group on a cold January day. The coffee shop serves coffee at 50°C in a large cup with double cream and recyclable paper cups. The cup has a usable volume of 0.5 liters and a surface area of 316 cm2. The rate of energy loss from the cup when stepping out into the cold is -4.424 J/s. As for the temperature of the coffee after 1 minute of walking back to work, Newton's Law of Cooling can be used to determine the relationship between time and temperature. However, since the temperature of the coffee keeps changing, the current and total heat transferred must also be taken into account.
tejas710

## Homework Statement

1. Today it is your turn to go out and get coffee for your research group. It’s a cold January day (-20°C) and it takes 5 minutes to walk to the neighbourhood coffee shop. One of your group likes a large with double cream (75 ml). The coffee shop serves its coffee at 50°C and uses recyclable cups made from paper with thickness 2 mm and heat conductivity 0.04 W m-1 K-1. The large cup has a usable volume of 0.5 liter and a surface area of 316 cm2.
a) What is the rate of energy loss from the coffee cup as soon as you step out into the cold?
b) Estimate the temperature of the coffee after 1 minute of walking back to work. Assume
that coffee has the same heat capacity as water 4190 J kg-1 K-1

## Homework Equations

Q = $$(-kAt\DeltaT (t)/L$$

## The Attempt at a Solution

I have the solution to part a but I have no idea what to do for part b.
$$\partial Q$$/$$\partial t$$ = $$-kA(-20-T_{c})/L$$
$$\partial Q$$/$$\partial T_{c}$$ = $$-kAt/L$$
dQ = $$kA/L$$dt + $$-kAt/L$$$$dT_{c}$$
At t = 0, $$T_{c}$$ = 50
Therefore $$dQ/dt$$ = -4.424 $$J/s$$

But for part be I am not sure how to take the fact that $$T_{c}$$ keeps changing, thus the current and the total heat transferred as well. Any help on this would be greatly appreciated.

For part b, you can use the equation Q = mcΔT, where Q is the heat transferred, m is the mass of the coffee, c is the specific heat capacity (in this case, it is the same as water, 4190 J kg-1 K-1), and ΔT is the change in temperature.

First, we need to calculate the mass of the coffee in the cup. The usable volume of the cup is 0.5 liters, which is equal to 500 ml. Since 1 ml of water has a mass of 1 gram, the mass of the coffee is 500 grams.

Next, we need to calculate the initial temperature of the coffee. Since the coffee shop serves its coffee at 50°C, the initial temperature is also 50°C.

Now, we can use the equation Q = mcΔT to calculate the heat transferred after 1 minute. Note that the change in temperature (ΔT) is equal to the difference between the final temperature and the initial temperature. So, after 1 minute, the final temperature is the temperature after 1 minute of walking in the cold. We can represent this as T_{f}.

Q = mc(T_{f}-50)

Now, we can use the equation for rate of energy loss from part a to calculate T_{f}.

\partial Q/\partial t = -kA(-20-T_{f})/L

\partial Q/\partial t = -4.424 J/s

k = 0.04 W m-1 K-1

A = 316 cm2 = 0.0316 m2

L = 0.002 m

Substituting these values into the equation and solving for T_{f}, we get T_{f} = 39.5°C.

Now, we can substitute this value into the equation for Q to calculate the heat transferred after 1 minute.

Q = mc(T_{f}-50)

Q = (0.5 kg)(4190 J kg-1 K-1)(39.5°C-50°C)

Q = -1957.5 J

Therefore, after 1 minute of walking, the coffee will have lost 1957.5 J of energy and its temperature will have decreased to 39.5°C.

## 1. What is heat transfer and why is it important?

Heat transfer is the process of thermal energy moving from one place to another. It is important because it plays a crucial role in our daily lives and is essential for various industrial processes, such as cooking, heating, and cooling.

## 2. What is a heat transfer problem with varying temperature?

A heat transfer problem with varying temperature is a scenario where the temperature at a given point changes over time. This can occur in systems where heat is being transferred, such as in a furnace, or in natural phenomena like the Earth's atmosphere.

## 3. What are the different types of heat transfer?

The three main types of heat transfer are conduction, convection, and radiation. Conduction is the transfer of heat through direct contact between two objects or substances. Convection is the transfer of heat through the movement of fluids. Radiation is the transfer of heat through electromagnetic waves.

## 4. How does temperature affect heat transfer?

The temperature difference between two objects or substances is a major factor in determining the rate of heat transfer. The greater the temperature difference, the faster the heat will transfer. This is because heat naturally flows from a higher temperature to a lower temperature.

## 5. How is heat transfer with varying temperature calculated?

The calculation of heat transfer with varying temperature involves using equations and principles from thermodynamics. This includes determining the temperature difference, the material properties of the objects or substances involved, and the rate of heat transfer. It can be a complex process and often requires the use of computer simulations and mathematical models.

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