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