Question

In a heat exchanger, the hot gases enter with a temperature of 150 °C and leave at 75 °C. The cold fluid enters at 25 °C and leaves at 125 °C. The capacity ratio of the exchanger is:

• A. 0.5
• B. 0.65
• C. 0.75
• D. 1

Hint:

The capacity ratio is always between 0 and 1. It indicates the relative sizes of the heat capacity rates of the two fluids. A value of 0 indicates that one fluid has a very large heat capacity rate (isothermal), and a value of 1 indicates that the heat capacity rates of both fluids are equal.

Answer 1

The Correct Option is C

Step 1: Understand the definition of capacity ratio.
The capacity ratio \( C^ \) of a heat exchanger is defined as the ratio of the smaller heat capacity rate to the larger heat capacity rate: \[ C^ = \frac{C_{min}}{C_{max}}, \] where \( C_{min} = (\dot{m}c_p)_{min} \) and \( C_{max} = (\dot{m}c_p)_{max} \). \( \dot{m} \) is the mass flow rate and \( c_p \) is the specific heat at constant pressure. Step 2: Apply the energy balance equation to both the hot and cold fluids.
For a heat exchanger with no heat loss to the surroundings, the heat lost by the hot fluid is equal to the heat gained by the cold fluid: \[ Q_h = Q_c \] \[ (\dot{m}c_p)_h (T_{h,i} - T_{h,o}) = (\dot{m}c_p)_c (T_{c,o} - T_{c,i}) \] where:
\( T_{h,i} \) is the inlet temperature of the hot fluid (150 °C).
\( T_{h,o} \) is the outlet temperature of the hot fluid (75 °C).
\( T_{c,i} \) is the inlet temperature of the cold fluid (25 °C).
\( T_{c,o} \) is the outlet temperature of the cold fluid (125 °C).
\( (\dot{m}c_p)_h \) is the heat capacity rate of the hot fluid (\( C_h \)).
\( (\dot{m}c_p)_c \) is the heat capacity rate of the cold fluid (\( C_c \)).
Step 3: Substitute the given temperatures into the energy balance equation.
\[ C_h (150 - 75) = C_c (125 - 25) \] \[ C_h (75) = C_c (100) \] Step 4: Determine the ratio of the heat capacity rates.
\[ \frac{C_h}{C_c} = \frac{100}{75} = \frac{4}{3} \approx 1.33 \] or \[ \frac{C_c}{C_h} = \frac{75}{100} = \frac{3}{4} = 0.75 \] Step 5: Identify \( C_{min} \) and \( C_{max} \).
Comparing \( C_h \) and \( C_c \), we see that \( C_c<C_h \). Therefore: \[ C_{min} = C_c \] \[ C_{max} = C_h \] Step 6: Calculate the capacity ratio \( C^ \).
\[ C^ = \frac{C_{min}}{C_{max}} = \frac{C_c}{C_h} = 0.75 \] Step 7: Select the correct answer.
The capacity ratio of the heat exchanger is 0.75, which corresponds to option 3.