Determining the flow rate for a cooling system

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To determine the flow rate for a cooling system requiring 1 MW of cooling power, the user applies the formula P = mcΔT/t, where P is power, m is mass, c is specific heat, and ΔT is the temperature change. The user calculates the flow rate using the density of water (1000 kg/m³) and arrives at a flow rate of 0.0476 m³/s. There is uncertainty regarding the placement of the density in the equation, specifically whether it should be in the numerator or denominator. The user seeks validation of their approach and the reasonableness of the calculated flow rate. Clarification on the dimensional analysis and the correct application of the density is requested.
David0709
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I need help with the following question:

Please have a look at the question and my attempt at the solution.

Alternative cooling systems are considered for a large computing centre requiring 1 MW of cooling power.(i) One solution is to provide cooling by a heat exchanger interfaced to a cold waterpipe. The water, initially at 10◦C, is taken from a nearby lake.The water coming out of the heat exchanger has a temperature of 15◦C. Whatis the flow rate (in m3s−1)? The specific heat of water is 4200 J kg−1K−.

My attempt at a solution is as follows but uses a dimensional analysis argument and I am unsure whether this is is an appropriate method for tackling the question.

E = mcΔT

But Power = P = E/t = mcΔT/t

Therefore P/cΔT = m/t = Density *Volume /t

But we know density of water is 1000kg m^-3.

So we deduce that P/cΔT*1000 = V/t which has the correct dimensions.

The value I deduce is 0.0476m^3s-1 and am unsure whether that seems reasonable .
 
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I can't tell whether the 1000 is in the numerator or the denominator. Please use parenthesis.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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