Radiators

Calculate the radiator output if it has a thermal gradient (difference between inlet water and return temperatures)
a) 5°C
b) 10°C
c) 15°C
d) 20°C

A heating water volume flow is 45 kg/h. How fast the water flows from the supply pipe to the radiator

e) DN16 a
f) DN20?

The DN designation is the nominal diameter in millimeters. The tube inside diameter.

Correct answer:

P₁ =  583.3333 W
P₂ =  1166.6667 W
P₃ =  1750 W
P₄ =  2333.3333 W
v₁ =  13.8155 cm/s
v₂ =  8.8419 cm/s

Step-by-step explanation:

$$\begin{gathered} t_1=5\text{°C} \\ Q=100\text{kg/h} \\ T=1\text{h}\to \text{s}=1\cdot 3600\text{s}=3600\text{s} \\ c=4200\text{J/kg/°C} \\ W=Q\cdot c\cdot t_1=100\cdot 4200\cdot 5=2100000\text{J/h} \\ {P}_1=W/T=2100000/3600=583.3333\text{W} \end{gathered}$$

$P_2=2\cdot P_1=2\cdot 583.3333=1166.6667\text{W}$

$P_3=3\cdot P_1=3\cdot 583.3333=1750\text{W}$

$P_4=4\cdot P_1=4\cdot 583.3333=2333.3333\text{W}$

$$\begin{gathered} D_1=16\text{mm}\to \text{cm}=16:10\text{cm}=1.6\text{cm} \\ {S}_1=\pi \cdot (D_1/2{)}^2=3.1416\cdot (1.6/2{)}^2\doteq 2.0106{\text{cm}}^2 \\ \rho =1000{\text{kg/m}}^3\to {\text{g/cm}}^3=1000:1000{\text{g/cm}}^3=1{\text{g/cm}}^3 \\ {V}_1=Q/\rho =100/1=100{\text{dm}}^3\text{/h} \\ {V}_2=V_1\to {\text{cm}}^3=V_1\cdot 1000{\text{cm}}^3=100\cdot 1000{\text{cm}}^3=100000{\text{cm}}^3 \\ {V}_3=V_2/T=100000/3600=\frac{250}{9}\doteq 27.7778{\text{cm}}^3\text{/s} \\ {v}_1=V_3/S_1=27.7778/2.0106=13.8155\text{cm/s} \end{gathered}$$

$$\begin{gathered} D_2=20\text{mm}\to \text{cm}=20:10\text{cm}=2\text{cm} \\ {S}_2=\pi \cdot (D_2/2{)}^2=3.1416\cdot (2/2{)}^2\doteq 3.1416{\text{cm}}^2 \\ {v}_2=V_3/S_2=27.7778/3.1416=8.8419\text{cm/s} \end{gathered}$$

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