Question

What are the restrictions on the relation \(U A_{s}=\) \(U_{i} A_{i}=U_{o} A_{o}\) for a heat exchanger? Here \(A_{s}\) is the heat transfer surface area and \(U\) is the overall heat transfer coefficient.

Short answer

Answer: The main restrictions on the relation \(U A_{s}= U_{i} A_{i}=U_{o} A_{o}\) in a heat exchanger are: 1. The total heat transfer surface area must be equal to the sum of the inner and outer heat transfer surface areas (\(A_{s} = A_{i} + A_{o}\)). 2. The heat transfer capacity between the fluids must be equal (\(U_{i}A_{i} = U_{o}A_{o}\)). These restrictions ensure that the heat exchanger operates without any temperature imbalances and with optimal heat transfer between the fluids.

Step-by-step solution

Understand the Variables

In a heat exchanger, the following variables are given: - \(U\): overall heat transfer coefficient - \(A_{s}\): heat transfer surface area - \(U_{i}\): overall heat transfer coefficient on the inner side (fluid side) - \(A_{i}\): heat transfer surface area on the inner side (fluid side) - \(U_{o}\): overall heat transfer coefficient on the outer side (shell side) - \(A_{o}\): heat transfer surface area on the outer side (shell side) The relation we are given is \(U A_{s}= U_{i} A_{i}=U_{o} A_{o}\), which describes the balance of heat transfer rates between the two fluids in the heat exchanger.

Consider the Constraint on Surface Areas

Firstly, we know the total heat transfer surface area depends on the inner surface area and the outer surface area: \(A_{s} = A_{i} + A_{o}\) This means that \(A_{s}\) is restricted by the sum of \(A_i\) and \(A_o\).

Consider the Constraint on Fluid Capacities

Another constraint we have to consider is that the heat transfer capacity of the fluids within the heat exchanger must be equal. If not, we would not have an effective heat exchanger as there would be a temperature imbalance. This constraint can be expressed as: \(U_{i}A_{i} = U_{o}A_{o}\)

Analyzing Constraints and Restrictions

By analyzing steps 2 and 3, we can see that there are two main restrictions on the relation \(U A_{s}= U_{i} A_{i}=U_{o} A_{o}\): 1. The total heat transfer surface area must be equal to the sum of the inner and outer heat transfer surface areas (\(A_{s} = A_{i} + A_{o}\)). 2. The heat transfer capacity between the fluids must be equal (\(U_{i}A_{i} = U_{o}A_{o}\)). These restrictions ensure that the heat exchanger operates without any temperature imbalances and with optimal heat transfer between the fluids.