Cold water (cp = 4180 j/kg·k) leading to a shower enters a thinwalled double-pipe counterflow heat exchanger at 15°c at a rate of 1.25 kg/s and is heated to 60°c by hot water (cp = 4190 j/kg·k) that enters at 100°c at a rate of 4 kg/s. if the overall heat transfer coefficient is 880 w/m2 ·k, determine the rate of heat transfer and the heat transfer surface area of the heat exchanger.

the rate of heat transfer Q is Q =235.125 kJ/s

the heat transfer surface area A of the heat exchanger is A= 15.30 m²

Explanation:

Assuming negligible loss  to the environment, then the heat flow of the hot water goes entirely to the cold water

Denoting a as cold water and b as hot water , then

Q= Fᵃ* cpᵃ * ( T₂ᵃ - T₁ᵃ)

where

F= mass flow

cp = specific heat capacity at constant pressure

T₂= final temperature

T₁ = initial temperature

replacing values

Q = Fᵃ* cᵃ * ( T₂ᵃ - T₁ᵃ) =  1.25 kg/s* 4180 J/kg·K* ( 60°C-15°C) * 1 kJ/1000J= 235.125 kJ/s

if all there is no loss to the surroundings

Qᵃ + Qᵇ = Q surroundings = 0

Fᵃ* cpᵃ * ( T₂ᵃ - T₁ᵃ) +  Fᵇ* cpᵇ * ( T₂ᵇ - T₁ᵇ) = 0

T₂ᵇ = T₁ᵇ - [Fᵃ* cpᵃ /  (Fᵇ* cpᵇ)  ]* ( T₂ᵃ - T₁ᵃ)

replacing values

T₂ᵇ =100°C - [1.25 kg/s* 4180 J/kg·K/  (4 kg/s* 4190 J/kg·K)]* ( 60°C-15°C)

T₂ᵇ = 85.97 °C

the heat transfer surface of the heat exchanger is calculated through

Q = U*A* ΔTlm

where

U= overall heat transfer coefficient

A = heat transfer area of the heat exchanger

ΔTlm = (ΔTend - ΔTbeg)/ ln ( ΔTend - ΔTbeg)

ΔTbeg = temperature difference between the 2 streams at the inlet of the heat exchanger (  hot out - cold in) = 85.97 °C - 15°C = 70.97 °C

ΔTbeg = temperature difference between the 2 streams at the end of the heat exchanger ( hot in - cold out ) = 100°C - 60 °C = 40°C

then

ΔTlm = (ΔTend - ΔTbeg)/ ln ( ΔTend - ΔTbeg) =( 70.97 °C-  40°C)/ ln( 70.97°C/40°C) = 17.455 °C

ΔTlm = 17.455 °C

then

Q = U*A* ΔTlm

A = Q/(U*ΔTlm) = 235.125 kJ/s/(17.455 °C *880 W/m²*K) *1000 J/kJ = 15.30 m²

A= 15.30 m²

think its the third option

graph question answered .. you think better when you are relaxed and happy