Question

A counter-flow double-pipe heat exchanger is to heat water from 20ºC to 80ºC at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at 160ºC at a mass flow rate of 2 kg/s. The inner tube is thin-walled, and has a diameter of 1.5 cm. If the overall heat transfer coefficient of the heat exchanger is 640 W/(m2.ºC), determine the length of the heat exchanger required to achieve the desired heating using the effectiveness-NTU method. Take the specific heat of geothermal water to be 4.31 kJ/(kg.ºC) and that of water to be 4.18 kJ/(kg.ºC).

Answer 1

Here the cold fluid (denoted as subscript c) is water and hot fluid (denoted as subscript h) is geothermal water.

Given: Initial and final temperature of water respectively, t₁= 20⁰C and t₂= 80⁰C

Mass flow rate of water, _c=1.2 kg/s

Initial temperature of geothermal water, T₁=160⁰C

Mass flow rate of geothermal water, _h=2 kg/s

Overall heat transfer coefficient, U= 640 W/m²⁰C

Diameter of the tube, D=1.5 cm= 0.015 m

Specific heat of geothermal water, C_ph=4.31 kJ/kg⁰C

Specific heat of water, C_pc=4.18 kJ/kg⁰C

Step 1: Determine the heat capacity flow rate of hot and cold fluid to get the minimum value of specific heat flow rate.

Heat capacity flow rate of hot fluid, C_h is C_h= _hC_ph

C_h= 2 kg/s*4.31 kJ/kg⁰C=8.62 kJ/s⁰C

C_h= 8.62 kW/⁰C

Heat capacity flow rate of cold fluid, C_c is C_c= _cC_pc

C_c = 1.2 kg/s*4.18 kJ/kg⁰C= 5.016 kJ/s⁰C

C_c = 5.016 kW/⁰C

As C_c < C_h, the minimum and maximum value will be C_min= 5.016 kW/⁰C and C_max = 8.62 kW/⁰C respectively.

The capacity ratio, c=C_min/C_max= 5.016/8.62=0.5819=0.582

c = 0.582

Step 2: Calculate the maximum heat transfer rate and actual heat transfer rate.

Maximum heat transfer rate, is calculated using the expression

The actual heat transfer rate, is calculated using the expression

Step 3: Calculate Effectiveness of heat exchanger and NTU

The effectiveness of a heat exchanger is

  

For a counter flow double pipe heat exchanger, Number of Transfer Units, NTU is obtained from the effectiveness relation for heat exchanger

where c is the Capacity Ratio (Determined in earlier steps)

Step 4: Length of heat exchanger required to achieve the desired heating is

where L is the Length of the heat exchanger

D is the Diameter of the tube

A_s is the Heat Transfer Surface Area

A_s is obtained from the expression

Thus the length of the heat exchanger for the calculated heat transfer surface area is

Therefore, the length of heat exchanger required is L= 108.45 m which can be approximated to 109 m.