Help solving a 1st order ODE for internal energy change

Question

The question is the following:

You would like to have a cup of tea, so you need to heat up water. In this exercise, you do that by putting water (0.5 liter) in a cooker that is electrically heating up its content.

The water out of the tap is 20^oC. The question here is: how long does the cooker need to be switched in order for the water to reach a temperature of 80^oC?

Negelect the heat capacity of the cooker. Furthermore, ignore heat losses to the surroundings

The specific heat of water is 4200 J/kgK. The electric power of the cooker is 1kW.

I am trying to solve this by solving the ODE of dE/dt=mcp(T-To) but am getting stuck on how to properly manipulate this equation in order to solve for Temp.

I know there is a shorthand way to do this but it is important i understand how to do it with the diff eq.

Answer 1

The basic equation for the heating of an object of mass "m" in "steady state" conditions is Q = mCpΔT so a person could differentiate this to get dQ/dt = mCp(dT/dt). In your case, you are providing a "power" input (1kW) and power is energy/time so, it is the dQ/dt. IF you assume that the power of 1kW is provided instantaneously and continuously to the water and there is no loss to the environment (the water is insulted so there is "no heat loss"), then we can modify the equation as follows:

dQ/dt = mCp(dT/dt) and we let P = dQ/dT then:

P = mCp(dT/dt) and we let dT be (T_f - T_i) where T_f = the final temperature and T_i is the initial temperature then:

P/(mCp) = (T_f - T_i)/dt or

Δt = mCp(T_f - T_i)/P

Using m = ρV (density times volume) the equation becomes:

Δt = ρVCp(T_f - T_i)/P

The Cp for water is 4,200 J/kg°C. The density (ρ) for water is about 1 kg/L at that temperature. 1kW = 1000W. So plugging in the numbers we get:

Δt = (1)(0.5)(4200)(80 - 20)/1000 = 126 seconds so just a little over 2 minutes.

As you can see, this is not a differential equation because we are assuming the energy/time is constant. A more realistic problem would be to have a function P(t) for the input power and to add in the heat loss to the environment (Q = hAdT/dt) and then you would have a differential equation.

Answer 2

T-T0 = dE/dt / mcp

T = T0 + (dE/dt) /mcp

T*dt = T0*dt + dE/mcp

T*dt - T0*dt = dE/mcp

dt = dE/mcp /(T-T0)

t = integral (dE) / mcp(T-T0) => E1-E0 / mcp(T-T0)