Please solve it in full details if you know how to solve it

Please solve it in full details if you know how to solve it.

Water with a flow rate of 5200 Kg/min is flowing inside the tubes of a cross flow heat exchanger. Water makes one pass through the tubes and its inlet temperature is 20 degree C. Combustion gas at 450 degree C flowing at a rate of 2500 Kg/min outside the tubes. Tube inside and outside diameters are D_0=20 mm and D0=25 mm and its length is 2m. Tube bank has 2500 tubes. Use the following properties:. Water properties at an assumed average mean temperature: rho= 987 Kg/m^3, C_p=4182 J/(kgK), mu = 5.28 times 10^-4 Ns/m^2, k= 0.645 W/(mK), P_r= 3.42 Combustion Gas properties at an assumed average film temperature. rho= 0.696 Kg/m^3, C_P=1030 J/(kgK), v=3.88 times 10^-5 m^2/s, k=0.041 W(mK), P_r=0.7 a(40). If tubes are made of steel with k=70 W/(mK), h_i= 850 W/(m^2K)-and an outlet temperature of 65 degree C is desired for water, using NTU-Effectiveness method calculate the overall heat transfer coefficient U_0 and then calculate the outside heat transfer coefficient h_0. b(10).What is the average heat flux based on inside area of the tubes? What is the average temperature gradient in the water on the inside surface of a tube?

Solution

The volumetric flow rate in a heating system can be expressed as

q = h / (c_p dt)         (1)

where

q = volumetric flow rate

h = heat flow rate

c_p = specific heat capacity

= density

dt = temperature difference

This generic equation can be modified for the actual units - SI or imperial - and the liquids in use.

For water with temperature 60 ^oF flow rate can be expressed as

q = h (7.48 gal/ft³) / ((1 Btu/lbm.^oF) (62.34 lb/ft³) (60 min/h) dt)        

    = h / (500 dt)         (2)

where

q = water flow rate (gal/min)

h = heat flow rate (Btu/h)

= density (lb/ft³)

dt = temperature difference (^oF)

For more exact volumetric flow rates the properties of hot water should be used.

Water mass flow can be expressed as:

m = h / ((1.2 Btu/lbm.^oF) dt)        

    = h / (1.2 dt)     (3)

where

m = mass flow (lb_m/h)

Volumetric water flow in a heating system can be expressed with SI-units as

q = h / ((4.2 kJ/kg^oC) (1000 kg/m³) dt)        

    = h / (4200 dt)        (4)

where

q = water flow rate (m³/s)

h = heat flow rate (kW or kJ/s)

dt = temperature difference (^oC)

For more exact volumetric flow rates the properties of hot water should be used.

Mass flow of water can be expressed as:

m = h / ((4.2 kJ/kg.^oC) dt)

    = h / (4.2 dt)        (5)

where

m = mass flow rate (kg/s)

A water circulating heating systems delivers 230 kW with a temperature difference of 20^oC.

The volumetric flow can be calculated as:

q = (230 kW) / ((4.2 kJ/kg ^oC) (1000 kg/m³) (20 ^oC))

    = 2.7 10^-3 m³/s

The mass flow can be expressed as:

m = (230 kW) / ((4.2 kJ/kg ^oC) (20^oC))

    = 2.7 kg/s

10 liters of water is heated from 10^oC to 100^oC in 30 minutes. The heat flow rate can be calculated as

h = (4.2 kJ/kg ^oC) (1000 kg/m³) (10 liter) (1/1000 m³/liter) ((100^oC) - (10^oC)) / ((30 min) (60 s/min))

=  2.1 kJ/s (kW)

The 24V DC electric current required for the heating can be calculated as

I = (2.1 kW) (1000 W/kW)/ (24 V)

   = 87.5 Amps