In this exercise the students will (1) operate a tube-in-shell heat exchanger and (2) analyze heat- exchanger performance by the LMTD and -NTU methods.


A heat exchanger is a device in which energy is transferred from one fluid to another across a solid surface. Exchanger analysis and design therefore involve both convection and conduction. Radiative transfer between the exchanger and the environment can usually be neglected unless the exchanger is uninsulated and its external surfaces are very hot.

Two important problems in heat exchanger analysis are (1) rating existing heat exchangers and (ii) sizing heat exchangers for a particular application. Rating involves determination of the rate of heat transfer, the change in temperature of the two fluids, and the pressure drop across the heat exchanger. Sizing involves selection of a specific heat exchanger from those currently available or determining the dimensions for the design of a new heat exchanger, given the required rate of heat transfer and allowable pressure drop. The LMTD method can be readily used when the inlet and outlet temperatures of both the hot and cold fluids are known. When the outlet temperatures are not known, the LMTD can only be used in an iterative scheme. In this case the -NTU method can be used to simplify the analysis.

Energy Considerations

The first Law of Thermodynamics, in rate form, applied to a control volume (CV), can be expressed as


where  stands for mass-flow rate (e.g., 1bm/min or kg/min) crossing the CV boundaries, h is specific enthalpy (energy/mass),  surris the rate of heat transfer from the CV to its surroundings, and st is the rate of change of energy stored in the CV. This simplified form of the First Law assumes no work- producing processes, no energy generation inside the CV, and negligible kinetic and potential energy in the fluid streams entering and leaving the CV. In steady state operation the energy residing in the CV is constant, meaning that st=0. If, furthermore, the boundary of the CV is adiabatic (i.e., perfectly insulated), then surr =0. Under these circumstances Eq. (1) reduces to a simple balance of enthalpy inflow and enthalpy outflow: .


Applied to a heat exchanger with two streams passing through it, Eq. (2) can be rearranged to give

 h(hh,i-h h,o ) = c(hc,o-hc,i)

where the subscripts h and c indicate the hot and cold fluids, respectively, and i and o indicate inlet and outlet conditions. In words, Eq. (3) says that the rate of energy loss by the hot fluid (left-hand side) equals the rate of energy gain by the cold fluid. Remember: This rate balance holds only if the heat-exchanger envelope is adiabatic and the exchanger has reached a steady state.

Shell and Tube Heat Exchanger
Figure 1 is a schematic diagram of a shell-and-tube heat exchanger with one shell pass and one tube pass. The cross-counterflow mode of operation is indicated.

Figure 1. Shell-and-tube-heat exchanger with one shell pass and one tube pass; cross- counterflow operation.

Inside the heat exchanger the hot and cold fluid temperature distributions would have the form sketched in Fig. 2(a).

Figure 2. (a) Temperature distributions in a counterflow heat exchanger.

Figure 2. (b) Energy balance in a differential length element.

The points 1 and 2 on the x axis represent the two ends of the heat exchanger. Provided there is no energy loss to the environment and that the exchanger has reached steady state, then dq, the rate of heat transfer from the hot fluid, is exactly equal to the rate of heat transfer to the cold fluid in a differential length dx of the exchanger surface. For the special case of fluids that are not changing phase and have constant specific heats

dq = -h cp,h dTh= -C hdTh,


dq = c cp,c dTc = CcdTc,


where Ch and C care called the hot and cold fluid heat-capacity rates, respectively. Integration of Eqs (4) and (5) along the heat exchanger (from 1 to 2) gives

q = Ch(Th,i-Th,o)



q = Cc(Tc,o - Tc,i)


1. Logarithmic Mean Temperature Difference (LMTD) Method

The differential heat-transfer rate dq across the surface area element dA can also be expressed as

dq = UTdA,


where is the local temperature difference between the hot and cold fluids and U is the overall coefficient of heat transfer at dA. Both U andT vary with position inside the heat exchanger (i.e., x), but by combining Eqs (4) and (5) with Eq. (8) it is possible for a single pass exchanger to integrate over the exchanger contact surface from inlet to out. The result of the integration is

q = AUm  Tln,


where q is the total heat-transfer rate (BTU/min), A is the total internal contact area (ft2), Um is the mean overall coefficient of heat transfer (BTU/min ft2º F), defined as

and is the logarithmic mean temperature difference (LMTD), given by

As shown in Fig. 2(a),T1 = T h,i-Tc,o and T2 = Th,o-T c,i for the counterflow, single pass case. Equation (9) is also applied to more complicated heat-exchanger designs with multipass and cross- flow arrangements with a correction factor applied to the LMTD. See Ozisik (1). As mentioned above, if both inlet and outlet temperatures are specified the LMTD can be calculated from Eq. (11) and q from either Eq. (6) or Eq. (7). Then the product is given explicitly by Eq. (9). Further specification of then "sizes" the heat exchanger, i.e., determines A and the dimensions of the internal flow passages.

2. -NTU Method
In cases where only the inlet temperatures of the hot and cold fluids are known, the LMTD cannot be calculated beforehand and application of the LMTD method requires an iterative approach. The recommended approach is the effectiveness or -NTU method. The heat-exchanger effectiveness, , is defined by

= q/qmax,


where q is the actual rate of heat transfer from the hot to cold fluid, and qmax represents the maximum possible rate of heat transfer, which is given by the relation

qmax = Cmin(Th,i- Tc,i)


where Cmin is the smaller of the two heat capacity rates (see above, Eqs (4) and (5). Thus, the actual heat transfer rate can be expressed as

q =Cmin(Th,i - Tc,i)


and calculated, given the heat-exchanger effectiveness , the mass-flow rates and specific heats of the two fluids and the inlet temperatures.

The value of  depends on the heat-exchanger geometry and flow pattern (parallel flow, counterflow, cross flow, etc.). Theoretical relations for  and graphical characteristics are given by Ozisik (1) and Incropera & DeWitt (2) for a limited selection of heat-exchanger types. For a single pass counterflow exchanger like the one used in this exercise


where C Cmin / Cmax and N UmA / Cmin. The dimensionless factor is known as the number of transfer units . It is an indicator of the actual heat-transfer area or physical size of the exchanger. An experimental determination of effectiveness is found by


Figure A1 in the Appendix is a schematic diagram of the two flow loops which exchange energy through the heat exchanger. Hot water circulates through the exchanger shell while a chilled solution of propylene glycol (PG) in water (approximately 30% PG by weight) circulates through the tubes. The chilled water flow is driven by a constant speed centrifugal pump while the hot water flow comes from the building water supply. Both flows are controlled manually with valves. Mass flow rates are indicated by in-line rotameter-type flow meters. Calibration curves for the two flow meters are appended. Four thermocouples mounted close to the four ports of the heat exchanger and connected to a digital readout indicate T h,i,Th,o ,Tc,i , and Tc,o.
The principal geometrical characteristics of the heat exchanger are as follows:

Shell diameter (outer) 3.63 in. Shell length 27-1/4 in. Tube O.D. 0.250 in. Shell volume 0.70 gal. Tube volume(internal) 0.40 gal. Tube surface area 11.1 ft2 No. of tubes 76 Length of tubes 26-11/16 in.

1. Locate the calibration data plot and equation for the water flow meter in the lab write up.

2. Locate the calibration data plot and equation for the propylene glycol flow meter. Since this calibration was performed using water as the working fluid you must correct the equation for use with propylene glycol using the following relationship:

pg=H2 O(S.G.pg)1/2

The specific gravity (S.G.) of the propylene glycol is found on the graphics provided in the write- up for a 30% solution

3. Determine the flow meter readings for the sixteen experiments possible in the following array, choosing the water flow as the "minimum" fluid:

Table 1. Nominal Values of Fluid Heat Capacities for the Propylene Glycol Loop.
C(Cp)max  [Btu/min°F]

C is the ratio of the heat capacities of the two fluid streams which is defined by:

From the first row (5, 10, 20, 40) you can determine the water flow meter readings as well as the propylene glycol flow meter readings, i.e., C = 1.0. From the "C" ratio, determine the other propylene glycol flow meter readings for the sixteen experiments using the values of Cp from the graph provided in the write-up.

4. Execution of the experiment
a) Set the H2O flow meter at the lowest reading in the array and then monitor the difference between the inlet and outlet temperatures for both water and propylene glycol (C = 1.0) until a steady state is established (usually in a few minutes).

b) Measure and record the inlet, outlet and temperature difference for both the water and propylene glycol flows.

c) Change the propylene glycol flow to give C = .75, then .5 and .25, each time repeating a) and b) above.

d) Sequence through the second, third and fourth columns of the experimental array.

5. Lab report.
a) Organize your lab data and calculated values in a neat spreadsheet array. Use only the English system of units.

b) Plot the heat transfer to the propylene glycol vs. the log-mean-temperature difference.

c) From the sixteen experiments performed, determine the average overall heat transfer coefficient, U, from the following definition:

Heat Transfer = (Cp)PGTPG = U A TLMTD
Note that the slope of the curve plotted in b) is equal to UA.

d) On one plot, plot the effectiveness, , versus NTU and curve fit the data where C is a constant by making a plot similar to Fig. 11.15 of Incropera & DeWitt using eq. (15). Remember  from your data is determined from the four measured temperatures using eq. (16) and not from equation (15).

Appendix (Print out before coming to lab):
Fig A1:  Schematic diagram, flow loops
Fig A2: Calibration curve for PENWALT flowmeter No. YYN-3447 
Fig A3: Calibration curve for Fischer & Porter flowmeter No. M2-1014/55
Fig A4: Specific gravities of aqueous propylene glycol solutions.
Fig A5: Specific heats of aqueous propylene glycol solutions.

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