please help A system for heating water from an inlet temperature of T_m, r = 20

Question

please help

A system for heating water from an inlet temperature of T_m, r = 20 degree C to an outlet temperature of T_m _o = 60 degree C involves passing the water through a thick-walled tube, having inner and outer diameters of 20 mm and 40 mm respectively. The outer surface of the tube is well insulated, and electrical heating within the tube provides a uniform supply of heat at a rate of 10^6 W/m^3. For a water mass flow rate of 0.1 kg/s, how long must the tube be to achieve the desired outlet temperature? If the inner surface temperature of the tube T, = 70 degree C at the outlet, what is the local convection heat transfer coefficient at the outlet? Assume that c_p for water = 4.179 kJ/kg. degree C

Explanation / Answer

a)
For an insulated pipe, let di and do be inner and outer diameters respectively,
finding out the mean or average temperature in K
Tm = ( Tmi + Tio ) / 2 = (293.15 + 333.15 ) / 2 = 313.15 K
Just write down the energy balance,
Eaccumlated - Eout = 0 ( other terms will be 0)
This term will be equal m'Cp( Tmo - Tmi)
Eaccumlated =  m'Cp( Tmo - Tmi) = q * area of the pipe (q is given as 106 W/m3 in the question)
area of the pipe will be (d2/4)*L   
m'Cp( Tmo - Tmi) = (d2/4)*L
Cp = 4179 J/Kg K, substitute all the values and find out the length of the tube L to get the answer.

b) Let qo be the heat transfer rate at the outlet surface,
then qo = 1/4 * [ (do2 - di2) / di ] * q
therefore, the local heat transfer coeffcient at the outlet, ho at T = 70oC = 343.15 K
   ho = qo / (343.15 - Tmo )
plug in all values to get the correct answer.

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