Let's analyze and solve the problem step-by-step: --- # **Problem Summary** - Large rectangular storage tank with alternating layers of masonry slabs. - Each layer thickness: \( W = .05\,m \) - Material properties: - Density, \( \rho = 190\,kg/m^3 \) - Specific heat, \( c = 800\,J/(kg \cdot K) \) - Thermal conductivity, \( k = .70\,W/(m \cdot K) \) - Environmental conditions: - Outer surface temperature is maintained at \( T_{\infty} = 600^\circ C \) - Convection coefficient, \( h = 100\,W/(m^2 \cdot K) \) - Initial temperature of the masonry: \( 25^\circ C \) - Goal: - Find the time to reach 75% of maximum energy storage. - Find the maximum and minimum temperatures of the masonry at this time. --- # **Step 1: Understand the physical process** - The storage material absorbs heat from the hot gas passing through. - Heat transfer occurs via conduction through the layers and convection at the outer surface. - The goal is to find the time when the stored energy reaches 75% of its maximum. --- # **Step 2: Determine the maximum energy stored** Maximum energy stored, \( E_{max} \), is when the entire mass of the masonry reaches the temperature of the environment, i.e., \( T_{max} \approx T_{\infty} = 600^\circ C \). - Total volume of one layer: \[ V = W \times \text{area} \] Since the area isn't specified, we consider per unit area (say, 1 m²). The total mass per layer: \[ m = \rho \times W \times 1\,m^2 = 190 \times .05 = 95\,kg \] - The energy stored when heated from initial temperature \( T_i = 25^\circ C \) to \( T_{max} = 600^\circ C \): \[ E_{max} = m \times c \times (T_{max} - T_i) = 95 \times 800 \times (600 - 25) \] \[ E_{max} = 95 \times 800 \times 575 \] \[ E_{max} = 95 \times 800 \times 575 \approx 95 \times 460,000 = 43,700,000\,J \] --- # **Step 3: Determine the time to reach 75% of maximum energy** - 75% of \( E_{max} \): \[ E_{75\%} = .75 \times E_{max} \approx .75 \times 43,700,000 = 32,775,000\,J \] - Since heat transfer is dominated by conduction and convection, we model the process as a lumped system (assuming high Biot number, or to simplify). - The time to reach a certain temperature in a lumped system: \[ T(t) = T_{\infty} - [T_{\infty} - T_i] e^{-t/\tau} \] where \( \tau \) is the thermal time constant: \[ \tau = \frac{\rho c V}{hA} \] - \( V \) is the volume: \( V = W \times 1\,m^2 = .05\,m^3 \) - \( A \) is the surface area: for 1 m² cross-section, \( A = 2 \times (length + width) \), but since the length and width are not specified, we consider the surface area as 2 m² (assuming a simple slab). --- # **Step 4: Calculate the thermal time constant \( \tau \)** \[ \tau = \frac{\rho c V}{h A} \] - \( V = .05\,m^3 \) - \( A = 2\,m^2 \) \[ \tau = \frac{190 \times 800 \times .05}{100 \times 2} \] \[ \tau = \frac{190 \times 800 \times .05}{200} \] \[ \tau = \frac{190 \times 40}{200} = \frac{76,000}{200} = 380\,s \] --- # **Step 5: Find the time to reach 75% energy** - The temperature at time \( t \): \[ T(t) = T_{\infty} - (T_{\infty} - T_i) e^{-t/\tau} \] - The energy stored as a function of temperature: \[ E(t) \propto T(t) - T_i \] - To reach 75% energy: \[ \frac{E(t)}{E_{max}} = .75 \] - Since the energy is proportional to temperature difference: \[ \frac{T(t) - T_i}{T_{max} - T_i} = .75 \] \[ T(t) - 25 = .75 \times (600 - 25) = .75 \times 575 = 431.25 \] \[ T(t) = 25 + 431.25 = 456.25^\circ C \] - Solve for \( t \): \[ 456.25 = 600 - (600 - 25) e^{-t/380} \] \[ 456.25 = 600 - 575 e^{-t/380} \] \[ 575 e^{-t/380} = 600 - 456.25 = 143.75 \] \[ e^{-t/380} = \frac{143.75}{575} \approx .25 \] \[ -t/380 = \ln(.25) \approx -1.386 \] \[ t = 380 \times 1.386 \approx 526\,s \] --- # **Final Answers:** - **Time to reach 75% maximum energy storage:** approximately **526 seconds** (about 8.8 minutes). - **Maximum temperature of the masonry:** **600°C**. - **Minimum temperature of the masonry at this time:** **25°C** (initial temperature). --- This simplified analysis provides an approximate time for the energy storage process considering lumped capacity assumptions.