Temperature 7477

The pool with a length of l = 50 m and a width of s = 15 m has a depth of h1 = 1.2 m at the shallowest part of the wall. The depth then gradually increases to a depth of h2 = 1.5 m in the middle of the pool. = 4.5 m walls in the deepest part of the pool. Consider the specific heat capacity of water c = 4,200 J / (kg · °C).

a) What is the volume of water in the pool?
b) How long does it take to fill the pool with two inlets if each flows A = 5 liters / s?
c) How much GJ of heat is needed to heat the pool water by ∆t = 10 °C?
d) When filling the pool for the first time, we require that the water has a temperature of t = 25 °C.
How much hot water and how much cold water will we need if the temperature of cold water t1 = 10 °C and hot water t2 = 60 °C?

Correct answer:

V =  1631.25 m³
t =  45.3125 h
E =  68.5125 GJ
a =  1141.875 m³
b =  489.375 m³

Step-by-step explanation:

$$\begin{gathered} l=50\text{m} \\ s=15\text{m} \\ {h}_1=1.2\text{m} \\ {h}_2=1.5\text{m} \\ {h}_3=4.5\text{m} \\ c=4200\text{J/kg/°C} \\ {S}_1=(h_1+h_2)/2\cdot \frac{l}{2}=(1.2+1.5)/2\cdot \frac{50}{2}=\frac{135}{4}=33.75{\text{m}}^2 \\ {S}_2=(h_2+h_3)/2\cdot \frac{l}{2}=(1.5+4.5)/2\cdot \frac{50}{2}=75{\text{m}}^2 \\ V=s\cdot (S_1+S_2)=15\cdot (\frac{135}{4}+75)=\frac{6525}{4}{\text{m}}^3=1631.25{\text{m}}^3 \end{gathered}$$

$$\begin{gathered} V_1=V\to \text{l}=V\cdot 1000\text{l}=1631.25\cdot 1000\text{l}=1631250\text{l} \\ Q=5\text{l/s} \\ {t}_1=V_1/(2\cdot Q)=1631250/(2\cdot 5)=163125\text{s} \\ t=t_1\to \text{h}=t_1:3600\text{h}=163125:3600\text{h}=45.313\text{h}=\frac{725}{16}\text{h}=45.3125\text{h} \end{gathered}$$

$$\begin{gathered} \rho =1000{\text{kg/m}}^3 \\ m=\rho \cdot V=1000\cdot \frac{6525}{4}=\frac{1000\cdot 6525}{4}=\frac{6525000}{4}=1631250\text{kg} \\ T=10\text{°C} \\ {E}_1=m\cdot c\cdot T=1631250\cdot 4200\cdot 10=68512500000\text{J} \\ E=E_1/10^9=68512500000/10^9=\frac{5481}{80}\text{GJ}=68.5125\text{GJ} \end{gathered}$$

$$\begin{gathered} a+b=V \\ 10a+60b=25V \\ a+b=1631.25 \\ 10a+60b=25\cdot 1631.25 \\ a+b=1631.25 \\ 10\cdot a+60\cdot b=25\cdot 1631.25 \\ a+b=1631.25 \\ 10a+60b=40781.25 \\ Pivot:Row1\leftrightarrow Row2 \\ 10a+60b=40781.25 \\ a+b=1631.25 \\ Row2-\frac{1}{10}\cdot Row1\to Row2 \\ 10a+60b=40781.25 \\ -5b=-2446.88 \\ b=\frac{-2446.875}{-5}=489.375 \\ a=\frac{40781.25-60b}{10}=\frac{40781.25-60\cdot 489.375}{10}=1141.875=\frac{9135}{8}{\text{m}}^3=1141.875{\text{m}}^3 \\ a=\frac{9135}{8}=1141.875 \\ b=\frac{3915}{8}=489.375 \end{gathered}$$

$b=489.375=\frac{3915}{8}{\text{m}}^3=489.375{\text{m}}^3$

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