Answer:
The log-mean-temperature-difference is 24.03⁰C
Explanation:
First we need to know if the heat exchanger is in parallel flow or counter-flow. However, counter flow arrangement is best used to recover heat.
L.M.T.D for counter flow is given as;
![L.M.T.D =((T_h_f_1 -T_c_f_2)-(T_h_f_2 -T_c_f_1))/(2.3log[(T_h_f_1 -T_c_f_2)/(T_h_f_2 -T_c_f_1)])](https://img.qammunity.org/2021/formulas/mathematics/college/yypa6u68gd9ewsfvoa6ce0tuixve2buzr2.png)
where;
Thf₁ is the initial temperature of the hot fluid = 80°C
Tcf₂ is the final temperature of the cold fluid = 51.5°C
Thf₁ - Tcf₂ = 80 - 51.5 = 28.5⁰C
Thf₂ is the final temperature of the hot fluid = 30°C
Tcf₁ is the initial temperature of the cold fluid = 10°C
Thf₂ - Tcf₁ = 30 - 10 = 20⁰C
![L.M.T.D = (28.5 -20)/(2.3Log[(28.5)/(20)]) \\\\L.M.T.D = (8.5)/(0.3538) =24.03^oC](https://img.qammunity.org/2021/formulas/mathematics/college/evihwxhlsaiz3uicoa541fq5ugewma2zzi.png)
Therefore, the log-mean-temperature-difference is 24.03⁰C