Question

2. A counter flow tube-shell heat exchanger is used to heat a cold water stream from 18 to 78oC at a flow rate of 1 kg/s. Heating is provided by a superhot water stream in the shell at 160oC flowing at 1.8 kg/s. Inner tube diameter is 1.4 cm while the tube wall is very thin. Overall heat transfer coefficient based on the inner tube is 630 W/m2 K. Assume constant water properties in the cold stream; use values at 320K. The cp value for the hot stream is 4.30 kJ/kg K. (30%) a. Determine the length of the heat exchanger using the ε-NTU method; b. Estimate the cold side heat transfer coefficient.

Answer 1

a)
`L = 220\,m`, b)
`U_(o) \approx 0.63\,(kW)/(m^(2)\cdot ^(\textdegree)C)`

Step-by-step explanation:

a) The counterflow heat exchanger is presented in the attachment. Given that cold water is an uncompressible fluid, specific heat does not vary significantly with changes on temperature. Let assume that cold water has the following specific heat:

`c_(p,c) = 4.186\,(kJ)/(kg\cdot ^(\textdegree)C)`

The effectiveness of the counterflow heat exchanger as a function of the capacity ratio and NTU is:

`\epsilon = (1-e^(-NTU\cdot(1-c)))/(1-c\cdot e^(-NTU\cdot (1-c)))`

The capacity ratio is:

`c = (C_(min))/(C_(max))`

`c = ((1\,(kg)/(s) )\cdot(4.186\,(kW)/(kg^(\textdegree)C) ))/((1.8\,(kg)/(s) )\cdot(4.30\,(kW)/(kg^(\textdegree)C) ))`

`c = 0.541`

Heat exchangers with NTU greater than 3 have enormous heat transfer surfaces and are not justified economically. Let consider that
`NTU = 2.5`. The efectiveness of the heat exchanger is:

`\epsilon = (1-e^(-(2.5)\cdot(1-0.541)))/(1-(2.5)\cdot e^(-(2.5)\cdot (1-0.541)))`

`\epsilon \approx 0.824`

The real heat transfer rate is:

`\dot Q = \epsilon \cdot \dot Q_(max)`

`\dot Q = \epsilon \cdot C_(min)\cdot (T_(h,in)-T_(c,in))`

`\dot Q = (0.824)\cdot (4.186\,(kW)/(^(\textdegree)C) )\cdot (160^(\textdegree)C-18^(\textdegree)C)`

`\dot Q = 489.795\,kW`

The exit temperature of the hot fluid is:

`\dot Q = \dot m_(h)\cdot c_(p,h)\cdot (T_(h,in)-T_(h,out))`

`T_(h,out) = T_(h,in) - (\dot Q)/(\dot m_(h)\cdot c_(p,h))`

`T_(h,out) = 160^(\textdegree)C + (489.795\,kW)/((7.74\,(kW)/(^(\textdegree)C) ))`

`T_(h,out) = 96.719^(\textdegree)C`

The log mean temperature difference is determined herein:

`\Delta T_(lm) = ((T_(h,in)-T_(c, out))-(T_(h,out)-T_(c,in)))/(\ln(T_(h,in)-T_(c, out))/(T_(h,out)-T_(c,in)) )`

`\Delta T_(lm) = ((160^(\textdegree)C-78^(\textdegree)C)-(96.719^(\textdegree)C-18^(\textdegree)C))/(\ln(160^(\textdegree)C-78^(\textdegree)C)/(96.719^(\textdegree)C-18^(\textdegree)C) )`

`\Delta T_(lm) \approx 80.348^(\textdegree)C`

The heat transfer surface area is:

`A_(i) = (\dot Q)/(U_(i)\cdot \Delta T_(lm))`

`A_(i) = (489.795\,kW)/((0.63\,(kW)/(m^(2)\cdot ^(\textdegree)C) )\cdot(80.348^(\textdegree)C) )`

`A_(i) = 9.676\,m^(2)`

Length of a single pass counter flow heat exchanger is:

`L =(A_(i))/(\pi\cdot D_(i))`

`L = (9.676\,m^(2))/(\pi\cdot (0.014\,m))`

`L = 220\,m`

b) Given that tube wall is very thin, inner and outer heat transfer areas are similar and, consequently, the cold side heat transfer coefficient is approximately equal to the hot side heat transfer coefficient.

`U_(o) \approx 0.63\,(kW)/(m^(2)\cdot ^(\textdegree)C)`