Effectiveness and NTU MCQ Quiz - Objective Question with Answer for Effectiveness and NTU - Download Free PDF

Explanation:

Effectiveness of a Parallel Flow Heat Exchanger

Definition: In heat exchanger analysis, the effectiveness (ε) is a measure of how well a heat exchanger transfers heat relative to its maximum possible heat transfer. It is defined as the ratio of the actual heat transfer to the maximum possible heat transfer that could occur if the outlet temperature of one fluid reached the inlet temperature of the other fluid.

Mathematically, the effectiveness (ε) is expressed as:

ε = Q_actual / Q_max

Where:

• Q_actual: The actual heat transfer rate in the heat exchanger.
• Q_max: The maximum possible heat transfer rate, which occurs when one of the fluids undergoes the maximum possible temperature change.

For a parallel flow heat exchanger, the two fluids enter the heat exchanger at their respective inlet temperatures and flow in the same direction. The effectiveness of such a heat exchanger depends on the heat capacity ratio (C_r) and the number of transfer units (NTU).

Heat Capacity Ratio (C_r):

The heat capacity ratio is defined as:

C_r = C_min / C_max

Where:

• C_min: The smaller heat capacity rate of the two fluids (C = ṁ × c_p).
• C_max: The larger heat capacity rate of the two fluids.

When the heat capacity ratio (C_r) is equal to 1, it implies that both fluids have the same heat capacity rate.

Effectiveness for Parallel Flow Heat Exchanger:

The effectiveness of a parallel flow heat exchanger is given by the following equation:

ε = [1 - exp(-NTU × (1 + C_r))] / [1 + C_r]

Where:

• NTU: Number of Transfer Units, a dimensionless parameter that represents the size and effectiveness of the heat exchanger.
• C_r: Heat capacity ratio.

Analysis for C_r = 1:

When C_r = 1, the equation for effectiveness simplifies to:

ε = [1 - exp(-2 × NTU)] / 2

The maximum value of effectiveness occurs when NTU approaches infinity (i.e., a very large heat exchanger). In this case:

exp(-2 × NTU) → 0

Therefore, the effectiveness becomes:

ε = [1 - 0] / 2 = 0.5

Conclusion: For a parallel flow heat exchanger with a heat capacity ratio (C_r) of 1, the maximum value of effectiveness is 0.5.

Explanation:

The heat exchanger effectiveness is defined as the ratio of actual heat transfer to the maximum possible heat transfer.

\(\epsilon = \frac{{{\rm{Actual\;heat\;transfer}}}}{{{\rm{Maximum\;possible\;heat\;transfer}}}} = \frac{Q}{{{Q_{max}}}}\)

\(NTU = \frac{{UA}}{{{C_{min}}}}\)

NTU is a measure of the effectiveness of the heat exchanger.

The NTU is a measure of the heat transfer size of the exchanger; larger the value of NTU the closer the heat exchanger approaches its thermodynamic limit.

Capacity Ratio:

\(R = \frac{{{C_{min}}}}{{{C_{max}}}}\)

Effectiveness of a parallel flow heat exchanger:

\({\epsilon_{parallel}} = \frac{{1 - \exp \left[ { - NTU\left( {1 + R} \right)} \right]}}{{1 + R}}\)

Effectiveness of a counterflow heat exchanger:

\({\epsilon_{counter}} = \frac{{1 - \exp \left[ { - NTU\left( {1 - R} \right)} \right]}}{{1 - R\exp \left[ { - NTU\left( {1 - R} \right)} \right]}}\)

Therefore effectiveness is the function of both NTU and Capacity Ratio

Important Point:

For condensor and evaporator (R = 0)

In condenser and evaporators in which one fluid remains at constant temperature throughout the exchanger. Here Cmax = ∞ and thus \(R = \frac{{{C_{min}}}}{{{C_{max}}}} = 0\)

By using the above case we arrive at the following common expression for parallel flow as well as counter – flow heat exchangers.

\(\epsilon = 1 - {\rm{exp}}\left( { - NTU} \right)\)

Concept:

Heat exchanger:

A heat exchanger may be defined as the equipment that transfers the heat from hot fluid to cold fluid with maximum rate, minimum investment, and running cost.

Effectiveness(ϵ): Effectiveness is the ratio of actual heat transfer rate to the maximum possible heat transfer rate. 

Actual heat transfer = mhch (Th1 - Th2 ) = mccc (Tc2 - Tc1)

ϵ = \(\frac{{{{\dot Q}_{actual}}}}{{{{\dot Q}_{max}}}} = \frac{{{C_h}\left( {{T_{h1}} - {T_{h2}}} \right)}}{{{C_{min}}\left( {{T_{h1}} - {T_{c1}}} \right)}}\) = \(\frac{{{C_c}\left( {{T_{c2}}\; - \;{T_{c1}}} \right)}}{{{C_{min}}\left( {{T_{h1}}\; - \;{T_{c1}}} \right)}}\)

where

Ch = Heat capacity of hot fluid = mhch,

Cc = Heat capacity of cold fluid = mccc,

Cmin = Minimum of Ch and Cc,

Th1 = Inlet temperature of hot fluid, Th2 = Exit temperature of hot fluid, Tc1 = Inlet temperature of cold fluid, Tc2 = Exit temperature of cold fluid

Calculation:

Given:

C₁ = 1000 W/K

C₂ = 400 W/K = C_min

Effectiveness, ∈ = \(\rm \frac{q_{actual}}{q_{max}}\)

∈ = \(\rm \frac{q_{actual}}{C_{min}(T_{h_i}-T_{c_i})}=0.5\)

q_actual = 0.5 × 400 (350 - 300)

= 10000 watt = 10 kW

Explanation:-

• The Number of Transfer Units (NTU) Method is used to calculate the rate of heat transfer in heat exchangers (especially counter current exchangers) when there is insufficient information to calculate the Log­Mean Temperature Difference (LMTD).
• In heat exchanger analysis, if the fluid inlet and outlet temperatures are specified or can be determined by simple energy balance, the LMTD method can be used; but when these temperatures are not available The NTU or The Effectiveness method is used.

Difficulties with the LMTD method - 

• From the design equation:

Q = UA∆T_lm

• From energy balance:

Q = m_cC_pc (T_co − T_ci) = m_hC_ph(T_hi − T_ho)

• The LMTD method of heat exchanger design is difficult to use if we want to predict the performance of a heat exchanger.
• Here we would know: m_c, m_h, T_ci, T_hi, U, and A, however, we would not know: T_co or T_ho
• Hence, we cannot find: Q or ∆T_lm

To solve the above problem with the usual LMTD method:

1. We could guess a value for T_ho or T_co, find Q from a heat balance, and then from UA∆T_lm.
2. Using this Q value, find new values of T_ho and T_co.
3. We would need to progressively alter our guesses until the first and second-step values of T were equal.

This iterative method can readily be done by computer, but a direct method can also be used. This direct method is known as the Effectiveness-NTU method.

Explanation:

• NTU means a number of transfer units. It is a method to design the heat exchanger.

​\(NTU = \frac{{UA_s}}{{{C_{min}}}}\)

• The NTU is a measure of the heat transfer size of the exchanger; larger the value of NTU, the closer the heat exchanger approaches its thermodynamic limit.
• where U is the overall heat transfer coefficient and As is the heat transfer surface area of the heat exchanger.
• Note that NTU is proportional to As.
• Therefore for a given flow condition i.e. for specified values of U and Cmin, the values of NTU is a measure of the heat transfer surface area As.
• Thus the larger the NTU, the larger the heat exchanger.

Explanation:

Heat Exchanger:

It is the steady flow adiabatic open system in which two flowing fluids exchange or transfer heat between them, without losing or gaining any heat from the ambient.

In the heat exchanger, the rate of enthalpy decrease of hot fluid = The rate of enthalpy increase of cold fluid.

Using the energy balance equation\(\dot{m}_h\times c_{ph}\times{(T_{hi}\;-\;T_{he})}=\dot{m}_c\times c_{pc}\times{(T_{ce}\;-\;T_{ci})}\)

ṁ_h, ṁ_c = Mass flow rate of hot and cold fluid.

c_ph, c_pc = Specific heats of hot and cold fluid.

T_hi, T_ci = Hot and cold fluid inlet temperature, T_he,T_ce = Hot and cold fluid exit temperature.

Capacity rate ratio (C):

\(C = \frac{{{{\left( {\dot m{c_p}} \right)}_{smaller}}}}{{{{\left( {\dot m{c_p}} \right)}_{larger}}}}\),  (0 ≤ C ≤ 1)

C = 0, when one of the fluids undergoing phase change like in steam condenser or evaporator or boiler.

Number of transfer unit (NTU)

\(NTU = \frac{{UA}}{{{{\left( {\dot m{c_p}} \right)}_{smaller}}}}\), U = overall heat transfer coefficient, A = heat transfer area

The formula of effectiveness for:

Calculation:

Given:

NTU = 2 and C = 0 (since phase change)

\({\epsilon_{parallel}} = {\epsilon_{counter}} =1 - {e^{ - NTU}}= 1 - {e^{ - 2}}\)

Concept:

The heat exchanger effectiveness is defined as the ratio of actual heat transfer to the maximum possible heat transfer.

\(\epsilon = \frac{{{\rm{Actual\;heat\;transfer}}}}{{{\rm{Maximum\;possible\;heat\;transfer}}}} = \frac{Q}{{{Q_{max}}}}\)

\(NTU = \frac{{UA}}{{{C_{min}}}}\)

NTU is a measure of the effectiveness of the heat exchanger.

The NTU is a measure of the heat transfer size of the exchanger; larger the value of NTU, the closer the heat exchanger approaches its thermodynamic limit.

Capacity Ratio:

\(C_r = \frac{{{C_{min}}}}{{{C_{max}}}}\)

Effectiveness of a parallel flow heat exchanger:

\({\epsilon_{parallel}} = \frac{{1 - \exp \left[ { - NTU\left( {1 + C_r} \right)} \right]}}{{1 + C_r}}\)

Effectiveness of a counter-flow heat exchanger:

\({\epsilon_{counter}} = \frac{{1 - \exp \left[ { - NTU\left( {1 - C_r} \right)} \right]}}{{1 ~-~ C_r\exp \left[ { - NTU\left( {1 - C_r} \right)} \right]}}\)

For condenser and evaporator (C_r = 0)

In condenser and evaporators in which one fluid remains at a constant temperature throughout the exchanger. Here Cmax = ∞ and thus \(C_r = \frac{{{C_{min}}}}{{{C_{max}}}} = 0\)

By using the above case, we arrive at the following common expression for parallel flow as well as counter – flow heat exchangers.

\(\epsilon = 1 - {\rm{exp}}\left( { - NTU} \right)\)

Explanation:

The heat exchanger effectiveness is defined as the ratio of actual heat transfer to the maximum possible heat transfer.

\(ϵ = \frac{{{\rm{Actual\;heat\;transfer}}}}{{{\rm{Maximum\;possible\;heat\;transfer}}}} = \frac{Q}{{{Q_{max}}}}\)

\(NTU = \frac{{UA}}{{{C_{min}}}}\)

NTU is a measure of the effectiveness of the heat exchanger.

The NTU is a measure of the heat transfer size of the exchanger; larger the value of NTU, the closer the heat exchanger approaches its thermodynamic limit.

Capacity Ratio:

\(C_r = \frac{{{C_{min}}}}{{{C_{max}}}}\)

Effectiveness of a parallel flow heat exchanger:

\({ϵ_{parallel}} = \frac{{1 - \exp \left[ { - NTU\left( {1 + C_r} \right)} \right]}}{{1 + C_r}}\)

Effectiveness of a counter-flow heat exchanger:

\({ϵ_{counter}} = \frac{{1 - \exp \left[ { - NTU\left( {1 - C_r} \right)} \right]}}{{1 ~-~ C_r\exp \left[ { - NTU\left( {1 - C_r} \right)} \right]}}\)

For condenser and evaporator (Cr = 0)

In condenser and evaporators in which one fluid remains at a constant temperature throughout the exchanger. Here Cmax = ∞ and thus \(C_r = \frac{{{C_{min}}}}{{{C_{max}}}} = 0\)

By using the above case, we arrive at the following common expression for parallel flow as well as counter – flow heat exchangers.

ϵ =  1 -  exp(-NTU)

exp(-NTU) = 1 - ϵ

-NTU = ln (1 - ϵ)

NTU = -ln (1 - ϵ)

Concept: 

The heat capacity of the fluid is the product of mass flow rate and the specific heat of the fluid. 

The capacity ratio in the heat exchanger is the ratio of minimum heat capacity to the maximum heat capacity

Capacity Ratio \(R = \frac{{{C_{min}}}}{{{C_{max}}}}\)

In a heat exchanger  

Heat released by hot gas = Heat gained by cold gas

ṁ_h C_ph (T_hi - T_he) = ṁ_c C_pc (T_ce - T_ci)

Calculation:

Given, T_hi = 200°C, T_he = 150°C, T_ce = 140°C, T_ci = 40°C

Now,

ṁ_h C_ph (200 - 150) = ṁ_c C_pc (140 - 40)

ṁ_h C_ph × 50 = ṁ_c C_pc × 100

clearly cold fluid capacity is less compared to hot fluid

\(R = \frac{{50}}{{100}}= 0.5\)

Concept:

NTU is a measure of the effectiveness of the heat exchanger.

The NTU is a measure of the heat transfer size of the exchanger; larger the value of NTU, the closer the heat exchanger approaches its thermodynamic limit.

\(NTU = \frac{{UA}}{{{C_{min}}}}\)

C_min is the minimum heat capacity between the hot and cold fluid

Calculation:

Given:        

A = 50 m², U = 100 W/m² K,

C_max = C_min = 1000 W/K

Concept:

The heat exchanger effectiveness (ϵ) is defined as the ratio of actual heat transfer to the maximum possible heat transfer.

\(\epsilon = \frac{{{Q_{act}}}}{{{Q_{max}}}}\)

\(\epsilon = \frac{{{C_h}\left( {{T_{h1}} - {T_{h2}}} \right)}}{{{C_{min}}\left( {{T_{h1}} - {T_{c1}}} \right)}} = \frac{{{C_c}\left( {{T_{c2}} - {T_{c1}}} \right)}}{{{C_{min}}\left( {{T_{h1}} - {T_{c1}}} \right)}}\)

Qact = ṁhCph (Th1 – Th2) = ṁcĊpc (Tc2 – Tc1)

Fluid Capacity Rate, C : Ch = ṁhCph, Cc = ṁc Cpc

Qmax = C_min(Th1 – Tc1) 

Q = C_h(Th1 – Th2) = C_c(Tc2 – Tc1)

Q = Ch × 25 = Cc × 15

∴ Ch is less than Cc

The formula used will be:

\(\epsilon = \frac{{{C_h}\left( {{T_{h1}} - {T_{h2}}} \right)}}{{{C_{min}}\left( {{T_{h1}} - {T_{c1}}} \right)}} = \frac{{{}\left( {{T_{h1}} - {T_{h2}}} \right)}}{{{{}}\left( {{T_{h1}} - {T_{c1}}} \right)}}\)

\(\epsilon = \frac{{{}\left( {{65} - {{40}}} \right)}}{{{{}}\left( {{{65}} - {{15}}} \right)}}=0.5\)

Concept:

Heat exchanger effectiveness:

\(ϵ= \frac{{{{\rm{Q}}_{{\rm{actual}}}}}}{{{{\rm{Q}}_{{\rm{max}}}}}}\)

\({{\rm{Q}}_{{\rm{actul}}}} = {{\rm{\dot m}}_{\rm{h}}}{{\rm{c}}_{{\rm{ph}}}}\left( {{{\rm{T}}_{{\rm{h}}1}} - {{\rm{T}}_{{\rm{h}}2}}} \right) = {{\rm{\dot m}}_{\rm{c}}}{{\rm{c}}_{{\rm{pc}}}}\left( {{{\rm{T}}_{{\rm{c}}2}} - {{\rm{T}}_{{\rm{c}}1}}} \right)\)

\({{\rm{Q}}_{{\rm{max}}}} = {{\rm{C}}_{{\rm{min}}}}\left( {{{\rm{T}}_{{\rm{h}}1}} - {{\rm{T}}_{{\rm{h}}2}}} \right)\)

\(ϵ= \frac{{{{\rm{Q}}_{{\rm{actual}}}}}}{{{{\rm{Q}}_{{\rm{max}}}}}} = \frac{{{{\rm{C}}_{\rm{h}}}\left( {{{\rm{T}}_{{\rm{h}}1}} \;- \;{{\rm{T}}_{{\rm{h}}2}}} \right)}}{{{{\rm{C}}_{{\rm{min}}}}\left( {{{\rm{T}}_{{\rm{h}}1}} \;- \;{{\rm{T}}_{{\rm{c}}1}}} \right)}} = \frac{{{{\rm{C}}_{\rm{c}}}\left( {{{\rm{T}}_{{\rm{c}}2}} \;-\; {{\rm{T}}_{{\rm{c}}1}}} \right)}}{{{{\rm{C}}_{{\rm{min}}}}\left( {{{\rm{T}}_{{\rm{h}}1}} \;- \;{{\rm{T}}_{{\rm{c}}1}}} \right)}}\)

Calculation:

Given:

ϵ = 0.25, T_h1 = 200°C, T_h2 = 75°C, T_c1 = 40°C

Heat capacities of gases and air are not given, so for calculating cold fluid outlet temperature, we have to assume the heat capacity of cold air is minimum.

Because then only, the effectiveness formula includes cold air outlet temperature.

\(ϵ =\frac{\left( {{T}_{ce}}-{{T}_{ci}} \right)}{\left( {{T}_{hi}}-{{T}_{ci}} \right)}\)

\(0.25=\frac{{{T}_{ce}}-40}{200-40}\)

T_ce - 40 = 40

Explanation:

Overall heat transfer coefficient (OHTC):

• OHTC is defined as the quantity of rate of heat transfer takes place due to any mode per unit area for unit temperature difference.
• It doesn’t have any physical meaning, it’s an experimentally determined coefficient value.
• The concept of OHTC is valid for a steady-state. 
• In the case of a heat exchanger, there is a combined mode of heat transfer or multiple resistances present therefore we use,

​\(Q = \;UA{\bf{\Delta }}T \Rightarrow UA = \frac{1}{{{R_{Total}}}}\), 

where R Total is total resistance and U = OHTC

• Unit of ‘U’ is W/m2K
• U indicates the quantity of rate of heat transfer takes place due to any mode per unit area for the unit temperature difference.

Fouling factor indicates the resistance offered due to chemical deposits and scaling.

So, \(\frac{1}{{{U_{with\;fouling}}}} = \frac{1}{{{U_{without\;fouling}}}} + F\)

Calculation:

Given:

Uo = 400 W/m2K, \(F=\frac{1}{{2000}}\frac{m^2K}{{{}W}}\)

\(\begin{array}{l} \therefore \frac{1}{{{U_{with\;fouling}}}} = \frac{1}{{400}} + \frac{1}{{2000}}\\ \Rightarrow {U_{with\;fouling}} = 333\frac{W}{{{m^2}k}} \end{array}\)

Concept:

Effectiveness: It is the ratio of actual heat transfer to the maximum heat transfer.

\(\epsilon = \frac{Q_{actual}}{Q_{max}}\)

actual heat transfer, Q = C_h (T_h1 - T_h2) = Cc (T_c2 - Tc1) 

maximum heat transfer, Q_max = C_min (T_h1 - T_c1)

where, subscript 1 = inlet, 2 = exit, h = hot liquid, c = cold liquid

C = heat capacity, T = temperature

Calculation:

Given:

Th2 = 25 °C, Th1 = 37 °C

Tc1 = 4 °C, Tc2 = 18 °C

Using energy balance: to find C_min

Since Cmin is smaller one of Ch and Cc

Ch (Th1 - Th2) = Cc (Tc2 - Tc1)

Ch (37 - 25) = Cc (18 - 4)

Ch (12) = Cc (14)

we can see that Cc is less than Ch, hence

Cmin = C_c

\(\epsilon = \frac{C_{c}\ (T_{c2} - T_{c1})}{C_{min}\ (T_{h1} - T_{c1})}\)

\(\epsilon = \frac{(18^\circ - 4^\circ)}{ (37^\circ- 4^\circ)}\) = 14/33 = 0.42

Heat capacity

\({\dot C_h} = {\dot M_h}{C_{ph}} = 1 \times 4 = 4\frac{{kJ}}{{sec - k}}\)

And \({\dot C_h} = {\dot C_c}\)

Also effectiveness \(\epsilon = \frac{{1 - {e^{ - NTU\left( {1 + R} \right)}}}}{{1 + R}}\)             _____________(i)

Where \(R = \frac{{{C_c}}}{{{C_h}}} = \frac{{{C_{min}}}}{{{C_{max}}}} = 1\)

Also from energy balance \({C_h}\left( {{T_{{h_1}}} - {T_{{h_2}}}} \right) = {C_c}\left( {{T_c}_2 - {T_{{c_1}}}} \right)\)

\(\Rightarrow \frac{{{C_c}}}{{{C_h}}} = \frac{{{T_{{h_1}}} - {T_{{h_2}}}}}{{{T_{{c_2}}} - {T_{{c_1}}}}} = 1 \Rightarrow {T_{{h_1}}} - {T_{{h_2}}} = {T_{{c_2}}} - {T_{{c_1}}}\)

And \(NTU = \frac{{UA}}{{{C_{min}}}} = \frac{{1 \times 5}}{4} = 1.25\)

Then from equation (1), we get

\(\epsilon = \frac{{1 - {\rm{exp}}\left( { - 1.25 \times 2} \right)}}{2} = 0.46\)

The maximum possible heat transfer

\({Q_{max}} = {C_{min}}\left( {{T_h}_1 - {T_{{c_1}}}} \right) = 4\left( {102 - 15} \right) = 348\ kW\)

Actual transfer Q_act = ϵ.Q_max = 0.46 × 348

\(\begin{array}{l} \Rightarrow {C_c}\left( {{T_{{c_2}}} - {T_{{c_1}}}} \right) = 0.46 \times 348\\ \Rightarrow {T_{{c_2}}} = \frac{{0.46 \times 348}}{4} + 15 = 55^\circ C \end{array}\)