Question

Consider a counter-flow heat exchanger with the inlet temperatures of two fluids (1 and 2) being \(T_{1, in} = 300\) K and \(T_{2, in} = 350\) K. The heat capacity rates of the two fluids are \(C_1 = 1000\) W/K and \(C_2 = 400\) W/K, and the effectiveness of the heat exchanger is 0.5. The actual heat transfer rate is ............... kW. (Answer in integer)

Hint:

The effectiveness method is very direct. Just remember two steps: 1. Find \(C_{\min}\) from the two given heat capacity rates (\(C = \dot{m}c_p\)). 2. Calculate \(q_{\max} = C_{\min} \times (\text{largest possible temperature difference})\). 3. Then, \(q_{\text{actual}} = \epsilon \times q_{\max}\). This avoids calculating outlet temperatures or using the LMTD method.

Answer 1

Step 1: Understanding the Concept:
The problem asks for the actual heat transfer rate (\(q_{\text{actual}}\)) in a heat exchanger, given the inlet temperatures, heat capacity rates, and the effectiveness (\(\epsilon\)). The effectiveness method is a direct way to calculate this.
Step 2: Key Formula or Approach:
The effectiveness of a heat exchanger is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate. \[ \epsilon = \frac{q_{\text{actual}}}{q_{\max}} \] Therefore, the actual heat transfer rate can be calculated as: \[ q_{\text{actual}} = \epsilon \times q_{\max} \] The maximum possible heat transfer rate (\(q_{\max}\)) occurs when the fluid with the smaller heat capacity rate (\(C_{\min}\)) undergoes the maximum possible temperature change, which is the difference between the inlet temperatures of the hot and cold fluids. \[ q_{\max} = C_{\min} (T_{h,in} - T_{c,in}) \] Step 3: Detailed Calculation:
1. Identify Hot and Cold Fluids and their Inlet Temperatures: - Fluid 1 inlet: \(T_{1, in} = 300\) K
- Fluid 2 inlet: \(T_{2, in} = 350\) K
Since \(T_{2, in}>T_{1, in}\), Fluid 2 is the hot fluid and Fluid 1 is the cold fluid.
- \(T_{h,in} = 350\) K
- \(T_{c,in} = 300\) K
2. Identify Heat Capacity Rates and find \(C_{\min}\):
- \(C_1 = C_c = 1000\) W/K
- \(C_2 = C_h = 400\) W/K
Comparing the two, the minimum heat capacity rate is: \[ C_{\min} = C_2 = 400 \text{ W/K} \] 3. Calculate the Maximum Possible Heat Transfer Rate (\(q_{\max}\)): \[ q_{\max} = C_{\min} (T_{h,in} - T_{c,in}) \] \[ q_{\max} = 400 \text{ W/K} \times (350 \text{ K} - 300 \text{ K}) \] \[ q_{\max} = 400 \times 50 = 20000 \text{ W} = 20 \text{ kW} \] 4. Calculate the Actual Heat Transfer Rate (\(q_{\text{actual}}\)):
Given effectiveness, \(\epsilon = 0.5\). \[ q_{\text{actual}} = \epsilon \times q_{\max} = 0.5 \times 20 \text{ kW} = 10 \text{ kW} \] Step 4: Final Answer:
The actual heat transfer rate is 10 kW.
Step 5: Why This is Correct:
The solution correctly applies the effectiveness-NTU method formulas. It correctly identifies the minimum heat capacity rate and the maximum temperature difference to calculate \(q_{\max}\), and then uses the given effectiveness to find the actual heat transfer rate.