Heat Mass Transfer

DOI 10.1007/s00231-014-1455-5

ORIGINAL

Development of Nusselt number correlation using dimensional

analysis for plate heat exchanger with a carboxymethyl cellulose
solution
Karuppannan Muthamizhi · Ponnusamy Kalaichelvi 

Received: 23 July 2013 / Accepted: 3 November 2014

© Springer-Verlag Berlin Heidelberg 2014

Abstract  Versatile applications of plate heat exchangers d Port diameter (m)

(PHE’s) in various industrial processes signify their com- Cs Channel spacing (m)
mand over other types of heat exchangers. The objective of h Convective heat transfer coefficient (W/m2 K)
this work was to derive Nusselt number correlations using K Fluid thermal conductivity (W/m K)
dimensional analysis in terms of all the parameters to deter- m Mass flow rate (kg/s)
mine the heat transfer coefficients in a PHE for various NTU Number of transfer units
concentrations of carboxymethyl cellulose (CMC) solution NRe Reynolds number, dimensionless
and it was also compared with the available models in liter- NNu Nusselt number, dimensionless
ature. The heat transfer coefficient increases with increase NPr Prandtl number, dimensionless
in concentration of CMC from 0.1 to 0.6 %w/w and also NGr Grashoff number, dimensionless
increases with increase in mass flow rates of both cold and Q Rate of heat transfer (W)
hot fluids from 0.016 to 0.099 kg/s. The Nusselt number T Temperature (K)
correlation developed using dimensional analysis has pre- ΔT Temperature difference (K)
dicted the Nusselt number for the given PHE with a RMS Δx Plate thickness (m)
deviation of 14.61. Uexp Experimental overall heat transfer coefficient,
including correction factor (W/m2 K)
Abbreviations U Overall heat transfer coefficient (W/m2 K)
∗

PHE Plate heat exchanger v Velocity (m2/s)

CMC Carboxymethyl cellulose w Width of the plate (m)
RTD Resistance temperature detector k Consistency index (Pa sn)
n Flow behavior index
List of symbols K Plate thermal conductivity (W/m K)
Ap Effective plate heat transfer area (m2) y Dependent variable
c∗ Capacity ratio c∗ < 1, dimensionless a Coefficient of x
FT Log-mean temperature difference correction x Independent variable
factor, 0 < FT < 1 dimensionless b Constant
Nc Number of channels i, j and k Model parameters, dimensionless
Cp Specific heat of fluid at constant pressure (J/kg K)
Dh Hydraulic diameter (m) Greek symbols
l Plate length (m) βg Thermal expansion with acceleration due to
gravity (*m/s2 K)
ρ Density of the fluid (kg/m3)
K. Muthamizhi · P. Kalaichelvi (*)  ΔTlm Logarithmic mean temperature difference
Department of Chemical Engineering, National Institute
of Technology, Tiruchirappalli 620015, Tamil Nadu, India (LMTD) (K)
e-mail: kalai@nitt.edu ε Exchanger thermal effectiveness

13
 Heat Mass Transfer

Subscripts CMC, which is a non-Newtonian; a typical hydrocolloid

h Hot imparts a pronounced effect on gel formation, water reten-
c Cold tion, and emulsifying and aroma retention with no direct
i Fluid inlet influence on the taste and flavor of foodstuffs [4, 5]. CMC
o Fluid outlet finds its applications as a stabilizer, binder, thickener, sus-
ss Stainless steel pending and water-retaining agent, in ice-creams and fro-
max Maximum zen desserts, fluid and powdered fruit drinks, sauces and
creams, cake mixes and slimming and dietary foods [6].
Hence different concentrations of CMC solution have been
1 Introduction selected as working fluid for the present investigation.
Till date, a little research on the study of PHE with non-
Gasketed PHE’s are predominantly applied for liquid-to-liq- Newtonian fluids has been reported. Reilly and Tien [7] stud-
uid duty owing to their advantageous features such as very ied of natural convection heat transfer from heated vertical
compact in design, easy to dismount for maintenance, clean- plate to a non-Newtonian fluid. For water–water duty, heat
ing or for modification of the heat transfer area by the addi- transfer and pressure drop studies in PHE is presented in
tion or removal of plates [1]. The heat transfer surface area articles [8–11]. Afonso et al. [12] conducted an experimental
can be readily changed or rearranged due to the flexibility investigation to obtain a correlation for the determination of
of the number of plates, plate type, and pass arrangements. convective heat transfer coefficient in a PHE using yogurt as
The high turbulence due to plates reduces fouling to about test fluid. Based on the experimental studies of Afonso et al.
10–25 percent as that of a shell-and-tube exchanger. Indeed, [12, 13], Fernandes et al. [14–18] has demonstrated numeri-
leakage from one fluid to the other cannot take place unless cal studies on PHE with yogurt as test fluid. Similar numeri-
a plate develops a hole. Since the gasket is between the cal studies in a PHE using egg yolk as test fluid were con-
plates, any leakage from the gasket is to the outside of the ducted by Gut et al. [19]. Jokar et al. [20] investigated the
exchanger. Because of the high heat transfer coefficients, parameters that affect the two-phase heat transfer within the
reduced fouling, absence of bypass and leakage streams, and minichannel PHE’s using refrigerant R-134a and developed
pure counterflow arrangements, the surface area required for appropriate correlations using dimensional analysis. Carez-
a PHE is 1/3–1/2 that of a shell-and-tube exchanger for a zato et al. [21] has also investigated the non-Newtonian heat
given heat duty. This would reduce the cost, overall volume, transfer on a PHE with a generalized configuration, using
and maintenance space for the exchanger. Furthermore, the CMC as a test solution. Lin et al. [22] developed a dimen-
gross weight of a PHE is about only 1/6 as that of an equiva- sionless correlation to characterize the heat transfer perfor-
lent shell-and-tube exchanger. The residence time of fluid mance of the corrugated channel in a PHE using Bucking-
particles on a given side is approximately the same for uni- ham pi theorem. Warnakulasuriya et al. [23] has established
form heat applications such as sterilization, pasteurization, the correlation equations to predict the heat transfer and
and cooking. In case of expensive fluids, a faster transient pressure drop of an absorbent salt solution in a commer-
response and a better process control become inevitable. In cial PHE. Mahdi et al. [24] developed a two-dimensional
this regard, there could be no significant hot or cold spots dynamic fouling model for milk fouling in a PHE. Cabral
in the exchanger to avoid deterioration of heat sensitive flu- et al. [3] studied the thermo-physical (density) and flow
ids. PHE holds only a small volume of fluid with no hot or properties (rheology) of pineapple juice in a PHE over a con-
cold spots and thus is best suited for expensive fluids. High siderable range of temperature and soluble solid content to
thermal performance can be well achieved in PHE’s as the obtain a correlation for the determination of friction factor
high degree of counterflow in PHE’s makes even tempera- versus Reynolds number. Khan et al. [25] has been carried
ture approaches of up to 1 °C (2 °F) possible. Such a high out experimental heat transfer studies with water-water in
thermal effectiveness (up to about 93 percent) facilitates PHE for symmetric 30/30, 60/60, and mixed 30/60 chevron
economical low-grade heat recovery. Moreover, PHE’s angle plates and a correlation to estimate Nusselt number as
are devoid of flow-induced vibration, noise, high thermal a function of Reynolds number, Prandtl number and chev-
stresses, and entry impingement problems, normally exist- ron angle has been proposed. Pandey et al. [26] has investi-
ent in shell-and-tube heat exchangers [2]. gated effects of nanofluid (Al2O3 in water 2, 3 and 4 vol%)
For the food industry, heat transfer studies in the PHE’s and water as coolants on heat transfer, and pressure drop
with non-Newtonian fluid flow is of great importance, since and exergy losses in PHE and has derived empirical corre-
various processes such as, cooling and heating of milk, cit- lation for Nusselt number and friction factor for both water
rus juices, and tropical fruit pulp pasteurization and con- and nanofluid. Correlations for the Nusselt number and the
centration processes make use of PHE [3] and generally friction factor for 0.5 % aluminum oxide nanofluid in a PHE
food fluids shows non-Newtonian flow behavior. using Wilson plot method has developed by Dustin et al. [27]

13
Heat Mass Transfer

and they have reported that the addition of 0.5 % aluminum hot water storage tank, a cold fluid storage tank, immersed
oxide improve the convective and overall heat transfer coef- type of a pair of electrical heaters, a couple of liquid Rota
ficient as much as 11 and 4.85 % respectively. meters, resistance temperature detectors, a manometer, two
Very few articles are available on the performance monoblock pumps and separate collection tank for cold
analysis of PHE’s using non-Newtonian fluid for various fluid and hot water; which was recycled for reuse. A 25 litre
concentrations and different flow rates in the literature. capacity stainless steel tank for hot water storage was ther-
Researchers have used the dimensional analysis to develop mally well insulated to avoid heat loss to the atmosphere.
Nusselt number correlation for PHE using fluids other than Immersing type of electrical heaters of 2 kW capacity was
non-Newtonain fluid and for other geometries namely hori- fixed inside the hot water tank to raise the water tempera-
zontal plate using non-Newtonian fluid as working fluid. ture. A thermostat temperature controller with a range of
However, none has used dimensional analysis to arrive at 0–110 °C was connected with electrical heaters to set the
Nusselt number correlation for the combination of PHE temperature of hot water at a desired value.
and non-Newtonain fluid. Hence, the present study aimed Double pole on/off switch was connected with the 2 kW
to investigate the effects of flow rate and thermo-physical capacity electrical heaters. A monoblock type pump of
properties of CMC, a non-Newtonian fluid on the Nusselt 0.25 hp capacity was connected to the hot water storage
number in the PHE of specific configuration and to derive a tank to pump the hot water from the hot water storage tank
suitable correlation for Nusselt number from experimental to the PHE and a flow control valve in the same line was
data based on dimensional analysis. Least square method meant for regulating the flow. The cold fluid was stored in a
was applied to estimate the constants and the powers of the separate stainless steel tank of equal capacity and well con-
parameters involved in the developed correlation. Lastly, nected with another monoblock type pump of similar capac-
the developed Nusselt number correlation was validated ity used to pump cold fluid from cold fluid storage tank to
with the experimental data points, and also compared with PHE. A return flow line was provided to convey the cold
the available literature Nusselt number correlation. fluid discharged at the outlet back into the collecting tank.
Two liquid Rota meters with accuracy of ±2 % and meas-
urement range of 0–10 LPM were well connected separately
2 Materials and methods with the hot and the cold fluid lines to measure the fluid flow
rate. These liquid flow meters were calibrated within their
2.1 Experimental setup and procedure flow range. Four Resistance Temperature Detectors (RTDs)
of model type: PT 100 was mainly used to measure the inlet
A six-channel corrugated type PHE with different flow rates and outlet temperature of each fluid with an accuracy of
and concentrations of working fluid CMC was used in the ±0.1 °C. Out of the total four, two RTDs were separately
present study. The apparatus shown in Fig. 1 consisted of a placed at the inlet ports of both the fluids to measure the inlet

Fig. 1  Experimental setup of
Rota meter Collecting
PHE
(R1) tank 1
Stirrer 1
Hot
Tap
fluid
Water Thi
Heater 1
Hot fluid tank Tho
Flow
Stirrer 2 Pump control Tco Tci
1 valve 1
Power-law fluid
Cold Plate Heat
Flow fluid Exchanger
Pump
Heater 2 Control
2
Cold fluid valve 2 Rota meter (R2)
tank

Collecting
tank 2

13
 Heat Mass Transfer

fluid temperature and remaining two RTDs were separately Table 1  Specifications of PHE
placed at the outlet ports of both the fluids to measure the Parameter Value (m)
outlet fluid temperature. Digital temperature indicators with
channel selectors connected with RTDs displayed the output Plate thickness (Δx) 0.0008
results of RTDs. These RTDs were calibrated within their Plate width (w) 0.125
temperature range via the corresponding calibration proce- Plate length (l) 0.425
dure. Table 1 shows the specifications of PHE. Port diameter (d) 0.32
CMC pure (food grade) was furnished by M/s. Merck. Channel spacing (Cs) 0.004
For a constant hot fluid mass flow rate and concentration of
CMC, experimental data were collected for different flow
rates of CMC solution (0.016–0.099 kg/s). Likewise, all runs EXPERIMENTAL PLAN
were carried out for different concentrations (0.1–0.6 % w/w) COLD FLUID
and different hot fluid mass flow rates (0.016–0.099 kg/s) as HOT FLUID TEMPERATURE
FLOW RATE
FLOW RATE °C
detailed in the experimental plan shown in Fig. 2.
1 LPM 1 LPM
2 LPM 2 LPM
Thi
2.2 Data reduction for PHE with CMC
3 LPM 3 LPM
Tho
The heat load, Q, of a PHE, can be represented by Eq. (1a, 4 LPM 4 LPM Tci
b and c) 5 LPM 5 LPM
Tco
6 LPM 6 LPM
Qh = mh Cph (Thi − Tho ) (1a)

Qc = mc Cpc (Tco − Tci ) (1b) 0.1 - 0.6%w/w with the

WATER CMC interval of 0.1%w/w
Q = Ap Uexp Tlm FT (1c)
where Q = Qh = Qc, m is mass flow rate, Cp is specific heat Fig. 2  Experimental plan
of fluid, A is the effective PHE area, Uexp is the overall heat
transfer coefficient, ΔTlm is the logarithmic mean tempera-  
min(mh Cph , mc Cpc )
ture difference and FT is the correction factor [28]. C∗ = (4)
max(mh Cph , mc Cpc )
(Thi − Tco ) − (Tho − Tci )
�Tlm = (Thi −Tco ) (1d) Q
ln (Tho −Tci ) ε= (5)
Qmax
where Qmax = min(mh Cph , mc Cpc )(Thi − Tci )
Ap = lw (plate effective heat transfer area) (1e) FT is calculated using Eq. 2 incorporating Eqs. (3, 4 and
The correction factor FT is a function of the exchanger 5) in Eq. 2. Qh and Qc, heat load are calculated using Eq.
configuration number of transfer units (NTU), defined in (1a, b), then these values FT, ΔTlm, Q, Ap are used to calcu-
Eq. (3) and heat capacity ratio (c∗, defined in Eq. 4). For late Uexp values using Eq. (1c).
pure counter current flow ideal case, FT = 1. For all other After the determination of Uexp in Eq. (1c), the con-
types of flow distribution, 0 < FT < 1. For the most usual vective coefficient of the water side hh of the PHE was
configurations, researchers have utilized PHE simulation obtained using same flow analysis as described by Vla-
models to generate charts and tables in the form FT  =  FT sogiannis et al. [30]. Introducing pure water as the cold
(NTU, c∗) or ε = ε (NTU, c∗), where ε is the thermal effec- fluid and hot fluid as well, (hh  = hc) the exchanger was
tiveness of the exchanger [29]. first tested in a single-phase operation. To extract an accu-
    rate correlation for the hot fluid side heat transfer coef-
1 1−C ∗ ε
NTU(1−C ∗ ) ln 1−ε if c∗ < 1 ficient of the available PHE, a series of measurements
FT = ε (2) were analyzed by the variant of the modified Wilson plot
NTU−(1−ε) if c∗ > 1
technique.

(Nc − 1)Ap U ∗
 1 1 1 x
NTU = (3) Uexp
=
hh
+
hc
+
Kss (6)
min(mh Cph , mc Cpc )
Calculated hh was again substituted in Eq. 6 to predict
Q
where U ∗ = ATlm cold side heat transfer coefficient (hc) where Δx is the plate

13
Heat Mass Transfer

thickness and kss is the thermal conductivity of the plate dimensional analysis. The heat transfer rate (Q, W) per unit
material (stainless steel) [30]. area (A, m2) in PHE depends upon the parameters such as
The correlation whose general form presented in Eq. (7) Hydraulic diameter (Dh, m) of PHE, velocity (v, m/s), den-
is commonly used for Newtonian fluids under turbulent sity (ρ, kg/m3), specific heat (Cp, J/kg K), thermal expan-
flow, where Nusselt number, Reynolds number and Prandtl sion with gravitational force (βg, m/s/t), thermal conductiv-
number (Nu, Re and Pr respectively) are dimensionless ity (K, W/m K), flow behavior index (n), consistency index
numbers and i, j, and k are empirical parameters [2]. (k, Pa sn), temperature difference (ΔT, K) etc. of non-New-
tonian fluid. For dimensional analysis, the dependencies of
jk
NNu = iNRe NPr (7) these variables have been grouped together as follows
Q
A better additive has to be selected using the rheologi-

v, ρ, βg, ∆T , K, k, n, Cp , Dh = 0 (10)
cal parameters as the criteria for selection in order to best A
serve the food industry. Typical rheological parameters, In the present study, the number of variables involved
consistency index, which is a strong function of the con- in the process was more than the number of fundamental
centration of the solution and temperature, and flow index, dimensions, so the Buckingham method was considered.
which does not have a strong dependence on the concentra- This theorem states that the relationship between r variables
tion and temperature of the solution are used in the power- is expressed as a relationship between r  −  s non-dimen-
law model [31]. sional groups of variables (called π groups), where s is the
CMC commonly used as a stabilizer, thickener, gelling number of fundamental dimensions required to express the
agent or emulsifier due to its rheological properties could r variables. In Eq. (10), there were 10 variables present and
be well represented by the power-law model. Generalized all the variables were expressed in terms of the five basic
equations of Reynolds number and Prandtl number for non- fundamental dimensions (s), thus r = 10, s = 5. The num-
Newtonian fluid are presented in Eqs. (8 and 9) [32]. ber of π groups that were formed was r − s = 10 − 5 = 5
  but as n was already M0L0T0, dimensionless groups were
ρv2−n Dhn reduced to 4, f (π1, π2, π3, π4) = 0. As the π groups were
NRe = (8) all dimensionless, i.e. M0L0T0, the principle of dimensional
k
homogeneity was used to equate the dimensions for each
 � �n−1  π group. Hence, the following expression of heat transfer
v coefficient in terms of Nusselt number was obtained
 kCp Dh
NPr (9)

=
K
    
U cp �T βg
Nu = f n 1 2−2n 1−n n+2 2
D n−2 ρ n−2 D n−2 ρ n−2 K D n−2 ρ n−2
In these equations, NPr, ρ, v, Dh, K, NRe and Cp are
Prandtl Number, density, the average velocity of the fluid, (11)
hydraulic diameter, thermal conductivity, Reynolds Num- Equation (11) shows the relation between the Nusselt
ber and heat capacity respectively, and k and n are the rheo- number and all the possible variables that could affect heat
logical parameters, consistency index and power law index transfer in the PHE considered for the present study.
of the CMC solution respectively.

2.3 Development of Nusselt number correlation based 3 Results and discussion

on dimensional analysis
In the present study, a six-channel corrugated type PHE
Dimensional analysis is the mathematical technique of with working fluid CMC was used. The effects of concen-
deriving relations between physical quantities by identify- trations and flow rates of both the hot and cold fluids on
ing their dimensions. The dimension of any physical quan- Nusselt number were analyzed based on the experimental
tity is a combination of the basic physical dimensions that data, and were discussed below.
compose it. Length L, mass M, Temperature T, Heat Hand
time t are fundamental dimensions, all physical quantities 3.1 Effect of cold fluid mass flow rate on heat transfer
are expressed in terms of these fundamental dimensions. coefficient
Whenever it is possible to identify the factors involved in
a physical situation, dimensional analysis forms a rela- To study the effect of cold fluid mass flow rate on the
tionship between them. Manjula et al. [33] have devel- heat transfer coefficient obtained experimentally for PHE,
oped a correlation for mixing time of the jet mixer using values of heat transfer coefficient were plotted against

13
 Heat Mass Transfer

different flow rates of cold fluid at various concentrations Hot fluid mass flow rate, 0.099 kg/s
of CMC while maintaining a constant hot fluid flow rate of 70
CMC
0.099 kg/s, as shown in Fig. 3. Concentration
It is observed from Fig. 3 that the heat transfer coefficient
has increased with the increase in cold fluid mass flow rate 50 0.5 %w/w
from 0.016 to 0.099 kg/s as well as for 0.1, 0.3 and 0.5 %

NNu
w/w concentrations of CMC at a hot fluid mass flow rate of
0.3 %w/w
0.099 kg/s. The same was observed for 0.2, 0.4 and 0.6 % 30
w/w concentrations of CMC and 0.016–0.083 kg/s of hot
fluid mass flow rate. The increase in the values of the heat 0.1 %w/w
transfer coefficients in the aforementioned cases could be 10
attributed to the increase in flow rates as well as turbulence. 0.0565 0.057 0.0575 0.058
Thermo-physical properties like thermal conductivity, Cold fluid mass flow rate, kg/s
density, flow behaviour index, consistency index and spe-
cific heat depended on temperature and concentration of Fig. 3  The effect of cold fluid mass flow rate on heat transfer coef-
CMC solution. As the concentration of CMC increased, the ficient for 0.099 kg/s hot fluid mass flow rate
thermal conductivity of CMC decreased while, the ratio of
dry CMC mass to water quantity increased. This restricted CMC - 0.6 %w/w
the movement of the CMC solution and water, thus the 90
Cold fluid
ability of CMC to conduct heat, represented by thermal mass flow
conductivity also decreased. As carbohydrate CMC gran- rate
ule was the principal solid content of CMC its concentra- 70
tion directly affected the density of CMC. As a result of 0.083 kg/s
increase in concentration, mass of the CMC granules and
NNu

50
density of the CMC solution also increased [34].
An increase in the concentration of a dissolved or dis- 0.055 kg/s
persed substance of CMC solution generally gives rise to 30
an increased viscosity, as does increasing the molecular
0.016 kg/s
weight of a solute of CMC solution and the specific heat also
increases, when concentration increases. Congruently, it was 10
0.01 0.03 0.05 0.07 0.09
evident from the developed correlation that the heat transfer
coefficient was directly proportional to the properties such Hot fluid flow rate, kg/s
as density, specific heat and viscosity and inversely propor-
tional to thermal conductivity. This implied that the heat Fig. 4  The effect of hot fluid mass flow rate on the heat transfer coef-
ficient of 0.6 % w/w CMC
transfer coefficient would also increase with an increase in
concentration, validating the results shown in Fig. 3.
3.3 Estimation of powers in the Nusselt number
3.2 Effect of hot fluid mass flow rate on heat transfer correlation using experimental data
coefficient
LINEST (Excel tool function) was used to determine the
In order to analyze the effect of hot fluid mass flow rate on significance of the effect of each variable involved in the
the heat transfer coefficient, a graph was plotted between equation of Nusselt number. The LINEST function cal-
the heat transfer coefficient and the hot fluid flow rate for culates the statistics for a line by using the least squares
different cold fluid flow rates and a particular concentration method to calculate a straight line that best fits your data,
of CMC (0.6 % w/w), as illustrated in Fig. 4. and then returns an array that describes the line. Equa-
As the flow rate increases, driving force also increases, tion (11) can be rewritten by taking logarithm on both sides
so it is inferred from Fig. 4 that the convective heat trans- of the equation.
fer coefficient has increased with the increase in hot fluid    
mass flow rates from 0.016 to 0.099 kg/s, for 0.6 % w/w U cp
log Nu = a1 log n 1 + a2 log 2−2n 1−n
concentration of CMC and for cold fluid mass flow rate D n−2 ρ n−2 D n−2 ρ n−2 K
0.016, 0.055 and 0.083 kg/s. The same was observed for  
�T βg
0.1–0.5 % w/w concentrations of CMC and for 0.033, + a3 log n+2 2 +b (12)
0.066 and 0.099 kg/s cold fluid mass flow rate. D n−2 ρ n−2

13
Heat Mass Transfer

Table 2  Comparison of S. no. Correlation Working fluid and configuration Equation RMS deviation
experimental Nusselt number
of CMC-0.1 % w/w with 1. Developed correlation CMC 13 14.61
the literature correlation and No. of plates: 7
developed correlation
2. Afonso et al. [12] Yogurt 14 18.57
No. of plates: 15, 13, 11, 7 and 5
3. Fernandes et al. [16] Yogurt 15a 16.26
No. of plates: 15, 13, 11, 7 and 5 15b 26.03
15c 18.44

Equation (12) is in the form of y = ax + b, where y is 120

the dependent variable which is a function of the inde- +20% +10%
pendent variable, x is the independent variable, a is the 100
-10%
coefficient of x and b is a constant. In the present study,
80

Predicted NNu
Eq. (12) was fitted to the experimental data points. Least
square method has been used to calculate the coefficients of 60 -20%
the equation in Nusselt number which gives the best fit to
experimental data points. Equation (12) was reduced to the 40
following form:
� �0.651139 20 Eq. 13
U Developed
Nu = 0.415834 n 1 0
D n−2 ρ n−2 0 10 20 30 40 50 60 70
� �2.267862 � �−0.15291  Experimental NNu
cp �T βg
× 2−2n 1−n n+2 2

D n−2 ρ n−2 K D n−2 ρ n−2 Fig. 5  Experimental NNu versus predicted NNu using Eq. (10)
(13)
Root Mean Square (RMS) deviation between the experi-
mental Nusselt Number and Nusselt number estimated Nu = 1.808Re0.449 Pr 0.3 ; R2 = 0.987 (15c)
using the correlation (Eq. 13) was 14.61. Here R2 is coefficient of determination. This experimental
Nusselt number was compared with the Nusselt number
3.4 Comparison of the developed Nusselt number obtained using literature correlations as well as correla-
correlation with the Nusselt number correlation tion developed in the present study and the RMS deviations
available in literature have been presented in Table 2.
The RMS deviation calculated for Nusselt numbers
The experimental Nusselt number was calculated for PHE obtained using Afonso et al. [12] and Fernandes et al.
with CMC using Eqs. (1–6) and it was compared with the [16] correlations have shown more variations when com-
Nusselt number correlation developed by Afonso et al. [12] pared to the developed correlation as a result of usage of
(Eq.  14) and Fernandes et al. [16] (Eq. 15a, b, c). Model different configurations and fluids and Fig. 5 compares the
equations proposed by Afonso et al. [12] (Eq. 14) and experimental NNu with those predicted from Eq. (13) and
Fernandes et al. [16] are for stirred yogurt which is a non- RMS deviation varies between −20 and +20 %. From the
Newtonian fluid of shear-thinning type. Since CMC also variations of the exponents in Eqs. (6) and (13), it can be
comes under the same category, the experimental Nusselt inferred experimental NNu and predicted NNu are affected
number calculated in the present study are compared with by the factors listed in Eq. (10), respectively [22].
Nusselt number calculated using the Nusselt number cor-
relation proposed by Afonso et al. [12] (Eq. 14) and Fer- 3.5 Uncertainty analysis in PHE
nandes et al. [16].
The objective of well designed experiments is to mini-
Nu = 1.759Re0.455 Pr 0.3 (14) mize the error. Uncertainty is a measure of repeatability
of experiment and needed to prove the accuracy of the
Nu = 1.878Re0.463 Pr 0.3 ; R2 = 0.985 (15a)
experiments. The errors are based on the least counts and
Nu = 1.809Re0.347 Pr 0.3 ; R2 = 0.993 (15b) the sensitivities of the measuring instruments used in the

13
 Heat Mass Transfer

Table 3  The average possible error for the experimental parameters

S. no. Basic equation Uncertainty equation Uncertainty %

1.

ρv2−n Dh n
 2  2 0.5 1.98
Re Dh
NRe = k Re = Dh + mm

2.

1 1−C ∗ ε
    2  2  2  0.5 2.30
FT = NTU(1−C ∗ ) ln 1−ε if c∗ < 1 f
= Thi
+ Tho
+ Tci
+ Tco
f Thi Tho Tci Tco

3.
 2 0.5 0.41
ǫ = min m Cp ,mQCp
2  2 
( h h c c (Thi −Tci )) ǫ
ǫ =
Thi
Thi + T ho
Tho + T Tci
ci

4. 1 1 1 x
 2  2 0.5 1.78
Uexp 2
 
= + +
Uexp hc hh kss hc
hc = Uexp + h hh
h
+ x x

5. hc Dh
 2  2 0.5

1.75
Nu = K Nu hc
+ D h
Nu = hc Dh

present investigation. The detailed systematic error analysis financial support given for carrying out this investigation (Ref. No.
is made in the present study to estimate the error associated 22/514/10-EMR-II).
with experimentation as given Table 3.

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