# 3. Spherical Dewar for Storing Liquid Oxygen — Solution ## (a) Thermal Circuit and Analytical Expressions ### **Thermal Circuit Description** The heat transfer from the liquid oxygen to the surroundings occurs through several steps, each corresponding to a thermal resistance: 1. **Convection from liquid oxygen to inner shell:** Resistance: \( R_{\text{conv,in}} \) 2. **Conduction through metal shell:** Resistance: \( R_{\text{cond,metal}} \) 3. **Conduction through foam insulation:** Resistance: \( R_{\text{cond,ins}} \) 4. **Convection from foam to air:** Resistance: \( R_{\text{conv,out}} \) 5. **Radiation from foam to air:** Resistance: \( R_{\text{rad}} \) **Thermal circuit:** All resistances are in series, except convection and radiation at the outer surface, which are in parallel. #### **Thermal Circuit Diagram (Text Description):** ``` T_lo ----[R_conv,in]----[R_cond,metal]----[R_cond,ins]----*----> T_inf | [R_conv,out] | [R_rad] ``` *At the outer surface, \( R_{\text{conv,out}} \) and \( R_{\text{rad}} \) are in parallel.* --- ### **Analytical Expressions** Let: - \( r_i \): inner radius of metal shell - \( r_ \): outer radius of metal shell (also inner radius of insulation) - \( r_1 \): outer radius of insulation - \( k_m \): thermal conductivity of metal - \( k_{ins} \): thermal conductivity of insulation - \( h_{in} \): convective coefficient (liquid oxygen to inner shell) - \( h_{out} \): convective coefficient (foam to air) - \( h_{rad} \): radiation heat transfer coefficient (foam to air) - \( T_{lo} \): temperature of liquid oxygen - \( T_{\infty} \): temperature of surroundings #### 1. **Convection (liquid to inner shell):** \[ R_{\text{conv,in}} = \frac{1}{4\pi r_i^2 h_{in}} \] #### 2. **Conduction (metal shell, spherical):** \[ R_{\text{cond,metal}} = \frac{1}{4\pi k_m} \cdot \frac{r_ - r_i}{r_i r_} \] Or, more generally (for spherical shells): \[ R_{\text{cond,metal}} = \frac{1}{4\pi k_m} \left( \frac{1}{r_i} - \frac{1}{r_} \right) \] #### 3. **Conduction (foam insulation, spherical):** \[ R_{\text{cond,ins}} = \frac{1}{4\pi k_{ins}} \left( \frac{1}{r_} - \frac{1}{r_1} \right) \] #### 4. **Convection (foam to air):** \[ R_{\text{conv,out}} = \frac{1}{4\pi r_1^2 h_{out}} \] #### 5. **Radiation (foam to air):** \[ R_{\text{rad}} = \frac{1}{4\pi r_1^2 h_{rad}} \] where \( h_{rad} = 4\epsilon \sigma T_{avg}^3 \), and \( T_{avg} = \frac{T_{lo} + T_{\infty}}{2} \). --- ## (b) **Total Thermal Resistance \( R_{tot} \):** At the outer surface, convection and radiation are in parallel, so their *combined resistance* is: \[ \frac{1}{R_{\text{surf}}} = \frac{1}{R_{\text{conv,out}}} + \frac{1}{R_{\text{rad}}} \] \[ R_{\text{surf}} = \left( \frac{1}{R_{\text{conv,out}}} + \frac{1}{R_{\text{rad}}} \right)^{-1} \] ### **Total resistance is then:** \[ R_{tot} = R_{\text{conv,in}} + R_{\text{cond,metal}} + R_{\text{cond,ins}} + R_{\text{surf}} \] --- ### **Summary of the Analytical Expression** \[ \boxed{ R_{tot} = \frac{1}{4\pi r_i^2 h_{in}} + \frac{1}{4\pi k_m} \left( \frac{1}{r_i} - \frac{1}{r_} \right) + \frac{1}{4\pi k_{ins}} \left( \frac{1}{r_} - \frac{1}{r_1} \right) + \left[ \frac{1}{4\pi r_1^2 h_{out}} + \frac{1}{4\pi r_1^2 h_{rad}} \right]^{-1} } \] where - \( h_{rad} = 4 \epsilon \sigma T_{avg}^3 \) - \( T_{avg} = \frac{T_{lo} + T_{\infty}}{2} \) --- ## **Summary Table** | Resistance Type | Expression | |------------------------|-----------------------------------------------------------------------------| | **Convection (in)** | \( R_{\text{conv,in}} = \frac{1}{4\pi r_i^2 h_{in}} \) | | **Conduction (metal)** | \( R_{\text{cond,metal}} = \frac{1}{4\pi k_m}\left(\frac{1}{r_i} - \frac{1}{r_}\right) \) | | **Conduction (ins)** | \( R_{\text{cond,ins}} = \frac{1}{4\pi k_{ins}}\left(\frac{1}{r_} - \frac{1}{r_1}\right) \) | | **Convection (out)** | \( R_{\text{conv,out}} = \frac{1}{4\pi r_1^2 h_{out}} \) | | **Radiation** | \( R_{\text{rad}} = \frac{1}{4\pi r_1^2 h_{rad}} \) | --- **This completes the step-by-step solution.**

# 3. Spherical Dewar for Storing Liquid Oxygen — Solution ## (a) Thermal Circuit and Analytical Expressions ### **Thermal Circuit Description** The heat transfer from the liquid oxygen to the surroundings occurs through several steps, each corresponding to a thermal resistance: 1. **Convection from liquid oxygen to inner shell:** Resistance: \( R_{\text{conv,in}} \) 2. **Conduction through metal shell:** Resistance: \( R_{\text{cond,metal}} \) 3. **Conduction through foam insulation:** Resistance: \( R_{\text{cond,ins}} \) 4. **Convection from foam to air:** Resistance: \( R_{\text{conv,out}} \) 5. **Radiation from foam to air:** Resistance: \( R_{\text{rad}} \) **Thermal circuit:** All resistances are in series, except convection and radiation at the outer surface, which are in parallel. #### **Thermal Circuit Diagram (Text Description):** ``` T_lo ----[R_conv,in]----[R_cond,metal]----[R_cond,ins]----*----> T_inf | [R_conv,out] | [R_rad] ``` *At the outer surface, \( R_{\text{conv,out}} \) and \( R_{\text{rad}} \) are in parallel.* --- ### **Analytical Expressions** Let: - \( r_i \): inner radius of metal shell - \( r_o \): outer radius of metal shell (also inner radius of insulation) - \( r_1 \): outer radius of insulation - \( k_m \): thermal conductivity of metal - \( k_{ins} \): thermal conductivity of insulation - \( h_{in} \): convective coefficient (liquid oxygen to inner shell) - \( h_{out} \): convective coefficient (foam to air) - \( h_{rad} \): radiation heat transfer coefficient (foam to air) - \( T_{lo} \): temperature of liquid oxygen - \( T_{\infty} \): temperature of surroundings #### 1. **Convection (liquid to inner shell):** \[ R_{\text{conv,in}} = \frac{1}{4\pi r_i^2 h_{in}} \] #### 2. **Conduction (metal shell, spherical):** \[ R_{\text{cond,metal}} = \frac{1}{4\pi k_m} \left( \frac{1}{r_i} - \frac{1}{r_o} \right) \] #### 3. **Conduction (foam insulation, spherical):** \[ R_{\text{cond,ins}} = \frac{1}{4\pi k_{ins}} \left( \frac{1}{r_o} - \frac{1}{r_1} \right) \] #### 4. **Convection (foam to air):** \[ R_{\text{conv,out}} = \frac{1}{4\pi r_1^2 h_{out}} \] #### 5. **Radiation (foam to air):** \[ R_{\text{rad}} = \frac{1}{4\pi r_1^2 h_{rad}} \] where \( h_{rad} = 4\epsilon \sigma T_{avg}^3 \), and \( T_{avg} = \frac{T_{lo} + T_{\infty}}{2} \). --- ## (b) **Total Thermal Resistance \( R_{tot} \)** At the outer surface, convection and radiation are in parallel, so their *combined resistance* is: \[ \frac{1}{R_{\text{surf}}} = \frac{1}{R_{\text{conv,out}}} + \frac{1}{R_{\text{rad}}} \] \[ R_{\text{surf}} = \left( \frac{1}{R_{\text{conv,out}}} + \frac{1}{R_{\text{rad}}} \right)^{-1} \] ### **Total resistance is then:** \[ R_{tot} = R_{\text{conv,in}} + R_{\text{cond,metal}} + R_{\text{cond,ins}} + R_{\text{surf}} \] --- ### **Summary of the Analytical Expression** \[ \boxed{ R_{tot} = \frac{1}{4\pi r_i^2 h_{in}} + \frac{1}{4\pi k_m} \left( \frac{1}{r_i} - \frac{1}{r_o} \right) + \frac{1}{4\pi k_{ins}} \left( \frac{1}{r_o} - \frac{1}{r_1} \right) + \left[ \frac{1}{4\pi r_1^2 h_{out}} + \frac{1}{4\pi r_1^2 h_{rad}} \right]^{-1} } \] where - \( h_{rad} = 4 \epsilon \sigma T_{avg}^3 \) - \( T_{avg} = \frac{T_{lo} + T_{\infty}}{2} \) --- ## **Summary Table** | Resistance Type | Expression | |------------------------|-----------------------------------------------------------------------------| | **Convection (in)** | \( R_{\text{conv,in}} = \frac{1}{4\pi r_i^2 h_{in}} \) | | **Conduction (metal)** | \( R_{\text{cond,metal}} = \frac{1}{4\pi k_m}\left(\frac{1}{r_i} - \frac{1}{r_o}\right) \) | | **Conduction (ins)** | \( R_{\text{cond,ins}} = \frac{1}{4\pi k_{ins}}\left(\frac{1}{r_o} - \frac{1}{r_1}\right) \) | | **Convection (out)** | \( R_{\text{conv,out}} = \frac{1}{4\pi r_1^2 h_{out}} \) | | **Radiation** | \( R_{\text{rad}} = \frac{1}{4\pi r_1^2 h_{rad}} \) | --- **This completes the step-by-step solution.**