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## Homework Statement

A 2kW immersion heater for heating water in a large tank is to be made from a straight

horizontal sheathed element of diameter 10mm.

*A sheathed element is an electrical resistance wire embedded in magnesium oxide powder and enclosed in a stainless steel tube, (there will be one in your kettle bent to a compact shape).*

The tank has a thermostat that controls the water temperature at 60'C. If the surface temperature of the sheathed element is not to exceed 100'C, calculate the length of tube required assuming that convective losses from the element can be determined by the non-dimensional equation:

[itex] Nu = 0.53(Gr * Pr)^{0.25}[/itex]

For water at 75'C take

μ = 0.3783 ×10-3kg m-1s-1

cp= 4.191 kJ kg-1C-1

ρ= 974.7 kg m-3

k = 0.6633 W m-1C-1

β= 0.619 ×10-3K-1

g=9.81 m s-2

## Homework Equations

[itex]\frac{dQ}{dt}=A*h*ΔT[/itex] newtons law of cooling (definition of heat transfer coefficient)

[itex]Re = \frac{ρVL}{μ}[/itex] Reynolds Number (dimensionless)

[itex]Pr = \frac{μCp}{K}[/itex] Prandtl Number (dimensionless)

[itex]Gr = \frac{βgρ^{2}ΔTL^{3}}{μ^2}[/itex] Grashof Number (dimensionless)

[itex]Nu = \frac{hL}{k}[/itex] Nusselt Number (dimensionless)

## The Attempt at a Solution

I know I have done the first part correctly, I have got the answer sheet right next to me. I just cannot see how the Grashof Number was calculated.

First Part. Calculate the length of the element needed to dissipate 2KW. (as a function of h)

from newton's law of cooling:

[itex]\frac{dQ}{dt}=A*h*ΔT[/itex]

[itex]\frac{dQ}{dt}=(pi*d*L)*h*ΔT[/itex]

[itex]L = \frac{2000}{0.01*pi*40*h}[/itex] This is just the above equation rearranged with the known constants put in.

This equation gives the length at which a 10mm diameter element would transfer 2KW to the water. (when there is a temperature difference of 40 degrees and a heat transfer coefficient of h)

Second Part. Now we just need to calculate/estimate a value for the heat transfer coefficient between the element and the water.

To do this you need to:

Calculate the nusselt number with the equation given in the question.

Use the nusselt number to calculate the heat transfer coefficient.

**4. my problem**

The only problem is that you need to know the length of the element to calculate the Grashof number, and then you need it again to calculate the heat transfer coefficient from the nusselt number. So you need the length to calculate the length.

The information given in the question is all I was given in my question, and the relevant equations are all the equations in my data book which I think are relevent. There are no more equations that tell me the Gr Number.

In the answer sheet my tutor has done the first part the the same way I did the first part. And then he has just wrote:

Pr=2.390

Gr=1.6125*10^6

Nu=23.48

h=1558

I just can not see how he has calculated the Gr Number. Please help!

Thank you.

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