To calculate the amount of water needed to cool down the skin of a combustion chamber, we need to determine the amount of heat that needs to be removed from the system. We can use the following formula:

Q = m x c x ΔT

where Q is the amount of heat that needs to be removed, m is the mass of the combustion chamber, c is the specific heat capacity of the material, and ΔT is the temperature difference between the initial and final temperature.

Assuming the combustion chamber is made of s235 steel, we can find the specific heat capacity of the material to be approximately 0.49 J/g·K. The mass of the combustion chamber can be calculated using the formula for the volume of a cylinder:

V = πr^2h

where r is the radius of the cylinder and h is the height. Substituting the values given, we get:

V = π(0.25 m)^2(0.6 m) = 0.029 m^3

The volume of the wall can be calculated using the same formula and subtracting the volume of the inner cylinder. The inner radius of the cylinder can be calculated by subtracting the thickness of the wall from the outer radius:

r = 0.25 m - 0.002 m = 0.248 m

V_wall = π(0.252 m^2 - 0.248^2 m^2)(0.6 m) = 0.003 m^3

The mass of the combustion chamber can be found by multiplying the volume by the density of s235 steel, which is approximately 7850 kg/m^3:

m = V x ρ = (0.029 m^3 + 0.003 m^3) x 7850 kg/m^3 = 245.7 kg

Now we can calculate the amount of heat that needs to be removed from the combustion chamber:

Q = m x c x ΔT = 245.7 kg x 0.49 J/g·K x (1000 K - 293 K) = 57745.4 kJ

Assuming the water used to cool down the combustion chamber is at room temperature (293 K) and it will absorb all the heat without changing its temperature, we can calculate the mass of water needed using the specific heat capacity of water, which is approximately 4.18 J/g·K:

m_water = Q / (c_water x ΔT) = 57745.4 kJ / (4.18 J/g·K x (1000 K - 293 K)) = 40.9 kg

Therefore, approximately 40.9 kg of water is needed to cool down the skin of the combustion chamber of the rocket engine after a 1-minute test at 1000K.
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