Two fluids A and B exchange heat in counterflow heat exchanger. Fluid A enters at 700 K and fluid B enters at 300 K. Mass flow rate of A & B is 1 kg/s. Specific heat of fluid A and B is 1000 J/kg K and 4000 J/kg K. Net effectiveness of heat exchange is 75%. Exit temperature of Fluid B is 

Concept:

Heat exchanger:

A heat exchanger may be defined as the equipment that transfers the heat from hot fluid to cold fluid with maximum rate, minimum investment, and running cost.

Effectiveness(ϵ): Effectiveness is the ratio of actual heat transfer rate to the maximum possible heat transfer rate. 

Actual heat transfer = mhch (Th1 - Th2 ) = mccc (Tc2 - Tc1)

ϵ = \(\frac{{{{\dot Q}_{actual}}}}{{{{\dot Q}_{max}}}} = \frac{{{C_h}\left( {{T_{h1}} - {T_{h2}}} \right)}}{{{C_{min}}\left( {{T_{h1}} - {T_{c1}}} \right)}}\) = \(\frac{{{C_c}\left( {{T_{c2}}\; - \;{T_{c1}}} \right)}}{{{C_{min}}\left( {{T_{h1}}\; - \;{T_{c1}}} \right)}}\)

where

Ch = Heat capacity of hot fluid = mhch,

Cc = Heat capacity of cold fluid = mccc,

Cmin = Minimum of Ch and Cc,

Th1 = Inlet temperature of hot fluid, Th2 = Exit temperature of hot fluid, Tc1 = Inlet temperature of cold fluid, Tc2 = Exit temperature of cold fluid

Calculation:

Given:

Th1 = 700 K, Tc1 = 300 K, c_h = 1000 J/kg K, c_c = 4000 J/kg K, m_h = m_c =  1 kg/s, ϵ = 75% = \(\frac{3}{4}\)

mhch (Th1 - Th2 ) = mccc (Tc2 - Tc1)

1 × 1000 × (700 - T_h2) = 1 × 4000 × (T_c2 - 300)

4 × T_c2 + T_h2 = 1900       .......(i)

As, mccc > mhch

Therefore Cmin = mhch = C_h

Effectiveness can be written in the form of actual heat transfer for the hot fluid and the maximum heat transfer, both the terms will contain mhch (as Cmin = mhch), therefore the effectiveness will be, 

ϵ = \(\frac{{{C_h}\left( {{T_{h1}} - {T_{h2}}} \right)}}{{{C_{min}}\left( {{T_{h1}} - {T_{c1}}} \right)}}\)

\(\frac{3}{4} = \frac{T_{h1}\;-\;T_{h2}}{T_{h1}\;-\;T_{c1}} \)

4 (T_h1 - T_h2) = 3 × 400

 Th2 = 400 K

From (i)

T_c2 = 375 K