Energy balance calculation is often done for designing a heat exchanger to determine operating parameters for hot and cold fluids such as - inlet/outlet temperatures and flow rates.

**Table of Content:**

1. Energy Balance Equation

2. Energy Balance Calculations

## Energy Balance Equation

The energy balance equation can be used to determine the heat transfer rate (Q) if the mass flow rates, specific heat capacities, and inlet and outlet temperatures of the two fluid streams are known. It can also be used to determine the outlet temperatures of the fluid streams if the heat transfer rate and the inlet temperatures are known.

For hot fluid side of a heat exchanger let,

m_{H} : mass flow rate of the hot fluid in kg/hr

Cp_{H} : mass heat capacity of the hot fluid in Joules/kg^{0}C

Ti_{H} and To_{H} : Respectively inlet and outlet temperatures on exchanger hot side in ^{0}C

m_{C} : mass flow rate of the cold fluid in kg/hr

Cp_{C} : mass heat capacity of the cold fluid in Joules/kg^{0}C

Ti_{C} and To_{C} : Respectively inlet and outlet temperatures on exchanger cold side in ^{0}C

Heat lost by the hot fluid = -Q = m_{H} × Cp_{H} × (To_{H} - Ti_{H}) … (1)

Heat gained by the cold side = Q = m_{C} × Cp_{C} × (To_{C} - Ti_{C}) … (2)

Comparing equations (1) and (2),

m_{H} × Cp_{H} × (Ti_{H} - To_{H}) = m_{C} × Cp_{C} × (To_{C} - Ti_{C}) … (3) (Heat balance equation)

This energy (heat) balance equation can be solved for one variable for any given case. Out of total six variables in the equation (3), five should be fixed to determine the unknown variable.

It should also be noted that the mass balance equation is already applied in this case to develop equation (3).

The fact that m_{H}in = m_{H}out = m_{H} and m_{C}in = m_{C}out = m_{C} is already considered while writing equations (1) and (2). Hence application of mass balance equation for heat exchanger does not present any new information here.

## Energy Balance Calculations

**Problem Statement: **A concentric counter-current doube-pipe heat exchnager is used to lubricating oil. Water is used as coolant. The inlet temperature of oil is 100°C and outlet temperature is 55°C. While the inlet temperature of water is 35°C and outlet temperature is 42°C. The mass flowrate of water in the exchanger is 0.1 kg/s. Calculate mass flow rate of water needed to maintain operating conditions.

**Solution:**

Ti_{H} = 100°C , To_{H} = 55°C

Ti_{C}= 35°C , To_{C} = 42°C

Cp_{H} (Oil) = 2131 Joules/kg^{0}C, Cp_{C} (Water) =4178 Joules/kg^{0}C

m_{H} = 0.1 kg/s

Heat lost by the hot fluid = -Q = m_{H} × Cp_{H} × (To_{H} - Ti_{H}) … (1)

Heat gained by the cold side = Q = m_{C} × Cp_{C} × (To_{C} - Ti_{C}) … (2)

From energy (heat) balance equation (equation 3) we can write,

m_{H} × Cp_{H} × (Ti_{H} - To_{H}) = m_{C} × Cp_{C} × (To_{C} - Ti_{C})

0.1 × 2131 × (100 - 55) = m_{C} × 4178 × (42-35)

m_{C} = 0.33 kg/s

So from energy balance calculation on heat exchnager, mass flow rate of water needed to maintain operating conditions is 0.33 kg/s.