- #1

Modrisco89

## Homework Statement

I need to find the time it takes to heat a vessel of water which is closed.

Here is what I have:

Specific heat capacity of water: 4184 J kg^-1 K^-1

Density of water: 1000 kg m^-3

Volume of water added and volume of vessel: 500 L or 0.5 m^3

Heat capacity of the vessel: 91000 J K^-1

Power rating of the element: 25 kW

This is just a rough time estimator for a program I'm designing

## Homework Equations

I'm thinking the formula to use without taking into account ambient temperature is:

t = C(ΔT)/power

t = time, C = to heat capacity, ΔT = final temp - initial temp and power = power rating of element.

## The Attempt at a Solution

So I calculate C = (1000)×(0.5)x(4184) + 91000

And I plug in what ever temperature difference and the power of the heating element and calculate the time that way!

Now I want take into consideration of the ambient temperature outside the tank so it will take longer to heat up the tank.

Im looking Newton's law of cooling and noticed the equation:

T(t) = Ta + (T0 - Ta)e^(t/t0)

T(t) is the temperature at time t

Ta is the ambient temperature

T0 is the temperature at time 0

t0 is the time constant in seconds

Can I take my original equation for t above I'm the relevant equation and sub it into t0? And then solve for t in the solved DE to calculate the time it would take to heat the vessel with regards to ambient temperature

Hope I've been clear