Certainly! Let's carefully analyze and solve the problem step-by-step. --- ## **Problem Summary:** - **Copper plate:** - Thickness \( t_{c} = 1\,mm = .001\,m \) - Thermal conductivity \( k_{c} = 401\,W/m \cdot K \) - **Epoxy boards (sandwiching the copper):** - Thickness of each epoxy board \( t_{e} = 7\,mm = .007\,m \) - Thermal conductivity \( k_{e} = .26\,W/m \cdot K \) - Size: \( 15\,cm \times 20\,cm \) - **Thermal contact conductance:** - On both sides of the copper plate: \( h_{c} = 600\,W/m^{2} \cdot K \) --- ## **Objective:** Calculate the **error** involved in the total thermal resistance if the thermal contact conductances are **ignored**. --- ## **Step 1: Understand the physical model** The heat transfer involves: - Contact resistances on both sides of the copper plate - Conduction resistance through the copper plate - Conduction resistance through the epoxy boards --- ## **Step 2: Calculate individual resistances** ### **a) Contact resistance on each side:** The contact resistance \( R_{contact} \) per side is: \[ R_{contact} = \frac{1}{h_{c} \times A} \] where: - \( h_{c} = 600\,W/m^{2} \cdot K \) - \( A = 15\,cm \times 20\,cm = .15\,m \times .20\,m = .03\,m^{2} \) Calculating: \[ R_{contact} = \frac{1}{600 \times .03} = \frac{1}{180} \approx .00556\,K/W \] Since there are **two** such contact resistances (one on each side), the **total contact resistance**: \[ R_{contact, total} = 2 \times .00556 = .01111\,K/W \] --- ### **b) Resistance through the copper plate** Thermal conduction resistance: \[ R_{copper} = \frac{t_{c}}{k_{c} \times A} \] Plugging in the values: \[ R_{copper} = \frac{.001}{401 \times .03} \approx \frac{.001}{12.03} \approx 8.32 \times 10^{-5}\,K/W \] This is **very small** compared to contact resistance, but we'll include it for completeness. --- ### **c) Resistance through the epoxy boards** For each epoxy board: \[ R_{epoxy} = \frac{t_{e}}{k_{e} \times A} \] Calculating: \[ R_{epoxy} = \frac{.007}{.26 \times .03} = \frac{.007}{.0078} \approx .897\,K/W \] Since there are **two** epoxy boards (above and below): \[ R_{epoxy, total} = 2 \times .897 = 1.794\,K/W \] --- ## **Step 3: Total thermal resistance (with contact conductance)** Total resistance: \[ R_{total} = R_{contact, total} + R_{copper} + R_{epoxy, total} \] \[ R_{total} \approx .01111 + 8.32 \times 10^{-5} + 1.794 \approx 1.805\,K/W \] --- ## **Step 4: Total thermal resistance (ignoring contact conductance)** If **we ignore the contact conductance**, the contact resistances are considered **negligible**. Then: \[ R_{total}^{(ignore)} = R_{copper} + R_{epoxy, total} \approx 8.32 \times 10^{-5} + 1.794 \approx 1.794\,K/W \] --- ## **Step 5: Calculate the error** The **error** involved in ignoring the contact conductance is: \[ \text{Error} = R_{total} - R_{total}^{(ignore)} = 1.805 - 1.794 = .011\,K/W \] This is approximately **.011 K/W**, which is about **.6%** of the total resistance. --- ## **Final Answer:** > **The error involved in ignoring the thermal contact conductance is approximately \(\boxed{.011\,K/W}\).**