Dane:

Q₀=100kW

t₁^s/t₂^s = 140/80 °C

t₂ⁱ/t₁ⁱ = 90/75 °C

$$\lambda_{\text{st}} = 53,15\frac{W}{m*K}$$

1. Q = 1, 15 * Q₀ = 1, 15 * 100 = 115 kW

2.

$$\overset{\overline{}}{t_{i}} = \frac{t_{2}^{i} + t_{1}^{i}}{2}$$

$$\overset{\overline{}}{t_{i}} = \frac{90 + 75}{2} = 82,5\ C$$

$$\rho_{i} = 970,2\ \frac{\text{kg}}{m^{3}}$$

$$c_{\text{pi}} = 4,198\frac{\text{kJ}}{kg*K}$$

$$\dot{V_{i}} = \frac{\dot{Q}}{c_{\text{pi}}*\rho_{i}*(t_{2}^{i} - t_{1}^{i})}$$

$$\dot{V_{i}} = \frac{115}{4,198*970,2*(90 - 75)} = 1,882*10^{- 3}\frac{m^{3}}{s}$$

3.

$$\frac{D_{n}}{d_{n}} = \frac{40}{20}\text{\ mm}$$

D_w = 0, 0425 m

d_z = 0, 0269 m

$$w_{i} = \frac{4*\dot{V_{i}}}{\pi*\left( D_{w}^{2} - d_{z}^{2} \right)}$$

$$w_{i} = \frac{4*1,882*10^{- 3}}{\pi*\left( {0,0425}^{2} - {0,0269}^{2} \right)} = 2,213\ \frac{m}{s}\ \in \left( 0,5\ ;2,5 \right)$$

4.

$$\dot{V_{s}} = \frac{\dot{Q}}{c_{\text{ps}}*\rho_{s}*(t_{2}^{s} - t_{1}^{s})}$$

$$\overset{\overline{}}{t_{s}} = \frac{t_{1}^{s} + t_{2}^{s}}{2}$$

$$\overset{\overline{}}{t_{s}} = \frac{140 + 80}{2} = 110\ C$$

$$c_{\text{ps}} = 4,233\frac{\text{kJ}}{kg*K}$$

$$\rho_{s} = 951,0\ \frac{\text{kg}}{m^{3}}$$

$$\dot{V_{i}} = \frac{115}{4,233*951,0*(140 - 80)} = 4,761*10^{- 4}\frac{m^{3}}{s}$$

5.

d_w = 0, 0222 m

$$w_{s} = \frac{4*\dot{V_{s}}}{\pi*d_{w}^{2}}$$

$$w_{s} = \frac{4*4,761*10^{- 4}}{\pi*{0,0222}^{2}} = 1,230\ \frac{m}{s}\ \in \left( 0,5\ ;2,5 \right)$$

6.

$$\tau_{s} = 100\ < \ \overset{\overline{}}{t_{s}} = 110$$

$$\lambda_{\text{fs}} = 68,5*10^{- 2}\frac{W}{m*K}$$

Pr_fs = 1, 60

Pr_ws = 1, 75

$$\nu_{\text{fs}} = 0,272*10^{- 6}\frac{m^{2}}{s}$$

$$\text{Re}_{s} = \frac{w_{s}*d_{w}}{\nu_{\text{fs}}}$$

$$\text{Re}_{s} = \frac{1,230*0,0222}{0,272*10^{- 6}} = 100390\ > 10000$$

$$\text{Nu}_{\text{fs}} = 0,021*\text{Re}_{\text{fs}}^{0,8}*\Pr_{\text{fs}}^{0,43}*\left( \frac{\Pr_{\text{fs}}}{\Pr_{\text{ws}}} \right)^{0,11}$$

$$\text{Nu}_{\text{fs}} = 0,021*100390^{0,8}*{1,60}^{0,43}*\left( \frac{1,60}{1,75} \right)^{0,11} = 255,3$$

$$\alpha_{s} = \frac{\text{Nu}_{\text{fs}}*\lambda_{\text{fs}}}{d_{w}}$$

$$\alpha_{s} = \frac{255,3*68,5*10^{- 2}}{0,0222} = 7878\ \frac{W}{m^{2}*K}$$

7.

$$\tau_{i} = 90\ > \ \overset{\overline{}}{t_{i}} = 82,5$$

$$\lambda_{\text{fi}} = 67,6*10^{- 2}\frac{W}{m*K}$$

Pr_fi = 2, 15

Pr_wi = 1, 95

$$\nu_{\text{fi}} = 0,355*10^{- 6}\frac{m^{2}}{s}$$

$$\text{Re}_{i} = \frac{w_{i}*D_{r}}{\nu_{\text{fi}}}$$

$$\text{Re}_{s} = \frac{2,213*0,0156}{0,355*10^{- 6}} = 97247\ > 10000$$

$$D_{r} = \frac{4*F}{U} = \frac{4*\frac{\pi}{4}*\left( D_{w}^{2} - d_{z}^{2} \right)}{\pi*\left( D_{w} + d_{z} \right)} = D_{w} - d_{z} = 0,0425 - 0,0269 = 0,0156$$

$$\text{Nu}_{\text{fi}} = 0,021*\text{Re}_{\text{fi}}^{0,8}*\Pr_{\text{fi}}^{0,43}*\left( \frac{\Pr_{\text{fi}}}{\Pr_{wi}} \right)^{0,25}$$

$$\text{Nu}_{\text{fi}} = 0,021*97247^{0,8}*{2,15}^{0,43}*\left( \frac{2,15}{1,95} \right)^{0,25} = 292,5$$

$$\alpha_{i} = \frac{\text{Nu}_{\text{fi}}*\lambda_{\text{fi}}}{D_{r}}$$

$$\alpha_{i} = \frac{292,5*67,6*10^{- 2}}{0,0156} = 12675\ \frac{W}{m^{2}*K}$$

8.

$$k_{L} = \frac{1}{\frac{1}{\pi*d_{w}*\alpha_{s}} + \frac{1}{2*\pi*\lambda_{\text{st}}}*\ln{\left( \frac{d_{z}}{d_{w}} \right) + \frac{1}{\pi*d_{z}*\alpha_{i}}}}$$

$$k_{L} = \frac{1}{\frac{1}{\pi*0,0222*7878} + \frac{1}{2*\pi*53,15}*\ln{\left( \frac{0,0269}{0,0222} \right) + \frac{1}{\pi*0,0269*12675}}} = 401,9\ \frac{W}{m*K}$$

9.

t₁ = t₁^s − t₂ⁱ

t₁ = 140 − 90 = 50 K

t₂ = t₂^s − t₁ⁱ

t₂ = 80 − 75 = 5 K

$$t_{\log} = \frac{t_{1} - t_{2}}{\ln\frac{t_{1}}{t_{2}}}$$

$$t_{\log} = \frac{50 - 5}{\ln\frac{50}{5}} = 19,54\ K$$

10.

q_L = k_L * t_log

$$q_{L} = 401,9*19,54 = 7853\ \frac{W}{m}$$

11.

$$\tau_{s}^{\text{rz}} = \overset{\overline{}}{t_{s}} - q_{L}*\frac{1}{\pi*d_{w}*\alpha_{s}}$$

$$\tau_{s}^{\text{rz}} = 110 - 7853*\frac{1}{\pi*0,0222*7878} = 95,71\ $$

$$\delta_{\tau}^{s} = \left| \frac{\tau_{s}^{\text{rz}} - \tau_{s}}{\tau_{s}} \right|*100\%$$

$$\delta_{\tau}^{s} = \left| \frac{95,71 - 100}{100} \right|*100\% = 4,29\%\ < 5\%\ O.K.$$

$$\tau_{i}^{\text{rz}} = \overset{\overline{}}{t_{i}} - q_{L}*\frac{1}{\pi*d_{z}*\alpha_{i}}$$

$$\tau_{i}^{\text{rz}} = 82,5 - 7853*\frac{1}{\pi*0,0269*12675} = 89,83\ $$

$$\delta_{\tau}^{i} = \left| \frac{\tau_{i}^{\text{rz}} - \tau_{i}}{\tau_{i}} \right|*100\%$$

$$\delta_{\tau}^{i} = \left| \frac{89,83 - 90}{90} \right|*100\% = 0,19\%\ < 5\%\ O.K.$$

12.

$$L = \frac{Q}{q_{L}}$$

$$L = \frac{115*10^{3}}{7853} = 14,64\ m$$