Question

The bulk or mixed temperature of a fluid flowing through a pipe or duct is defined as (a) \(T_{b}=\frac{1}{A_{c}} \int_{A} T d A_{c}\) (b) \(T_{b}=\frac{1}{\dot{m}} \int_{A} T \rho V d A_{c}\) (c) \(T_{b}=\frac{1}{m} \int_{A^{c}} h \rho V d A_{c}\) (d) \(T_{b}=\frac{1}{A_{c}} \int_{A} h d A_{c}\) (e) \(T_{b}=\frac{1}{\bar{j}} \int_{A} T \rho V d A_{c}\)

Short answer

Answer: The correct formula to calculate the bulk or mixed temperature of a fluid flowing through a pipe or duct is: \(T_{b}=\frac{1}{\dot{m}} \int_{A} T \rho V d A_{c}\).

Step-by-step solution

Understand the concept of bulk or mixed temperature

The bulk or mixed temperature of a fluid flowing through a pipe or duct is the temperature averaged across the cross-sectional area of the pipe, taking into account the variations of the temperature profile and the mass flow rates across the area.

Analyze formula (a) : \(T_{b}=\frac{1}{A_{c}} \int_{A} T d A_{c}\)

Formula (a) calculates the average temperature across the pipe's cross-sectional area by integrating the local temperature \(T\) with respect to the local area \(dA_{c}\), and then dividing by the total area \(A_{c}\) of the pipe cross-section. However, it doesn't consider the mass flow rates across the area, which is required for calculating the bulk temperature. Therefore, this formula is incorrect.

Analyze formula (b) : \(T_{b}=\frac{1}{\dot{m}} \int_{A} T \rho V d A_{c}\)

Formula (b) calculates the bulk temperature by integrating the product of local temperature \(T\), density \(\rho\), and velocity \(V\) with respect to the local area \(dA_{c}\) across the pipe's cross-sectional area. It then divides this by the total mass flow rate through the pipe (\(\dot{m}\)), which is correct for calculating the bulk temperature. Therefore, this formula is likely the correct one.

Analyze formula (c) : \(T_{b}=\frac{1}{m} \int_{A^{c}} h \rho V d A_{c}\)

Formula (c) calculates the average temperature using the product of the local enthalpy \(h\), density \(\rho\), and velocity \(V\), and then divides by the total mass flow rate inside the pipe (\(m\)). However, this formula doesn't directly involve the local temperature \(T\) of the fluid and is based on enthalpy. Therefore, this formula is incorrect.

Analyze formula (d) : \(T_{b}=\frac{1}{A_{c}} \int_{A} h d A_{c}\)

Formula (d) calculates an average by integrating the local enthalpy \(h\) with respect to the local area \(dA_{c}\), and then divides it by the total area \(A_{c}\) of the pipe cross-section. This formula doesn't directly involve the local temperature \(T\) and also doesn't consider mass flow rates. Therefore, this formula is incorrect.

Analyze formula (e) : \(T_{b}=\frac{1}{\bar{j}} \int_{A} T \rho V d A_{c}\)

Formula (e) calculates the bulk temperature using the product of local temperature \(T\), density \(\rho\), and velocity \(V\), which is integrated with respect to the local area \(dA_{c}\) across the pipe's cross-sectional area. However, it divides by the average specific mass flow rate \(\bar{j}\) instead of the overall mass flow rate (\(\dot{m}\)) through the pipe. Therefore, this formula is incorrect.

Conclusion

Based on the analysis of all the proposed formulas, the correct formula to calculate the bulk or mixed temperature of a fluid flowing through a pipe or duct is option (b): \(T_{b}=\frac{1}{\dot{m}} \int_{A} T \rho V d A_{c}\).